32
Spectrum of Commutative Banach Algebras and Amenable Locally Compact Groups by Z.Hu Windsor Mathematics Report WMSR#96-05 October, 1996
Spectrum of Commutative Banach Algebras and Amenable Locally Compact Groups
by
ZHu
Windsor Mathematics Report
WMSR96-05
October 1996
SPECTRUM OF COMMUTATIVE BANACH ALGEBRAS
AND AMENABLE LOCALLY COMPACT GROUPS
ZHIGUO Hu Department of Mathematics and Statistics
University of Windsor Windsor Ontario Canada N9B 3P4
Abstract Let A be a semisimple commutative regular tauberian Banach algebra
with spectrum EA We study the norm spectra of spanEA and present some applishy
cations to the Fourier algebra A(G) and the Figa-Talamanca-Herz algebra Ap( G)
of any locally compact group G The algebra A is said to have property (5) if
for all ltp E spanEA OJ ltp has a nonempty norm spectrum Let M~(G) denote
the Cmiddot-algebra generated by left translation operators on L2(G) and Gd denote
the discrete group G We prove that A(G) has property (5) iff the canonical trace
on M~(G) is faithful iff M~(G) ~ M~(~) In particular it is concluded that
Ll(G) has property (5) for all abelian groups G which generalizes the uniqueness
theorem on Ll(R) We further prove that Gd is amenable iff G is amenable and
Ap(G) has property (5) for some (and hence all) p
1991 Mathematics Subject Classification Primary 46J20 46J99 43A30 22D25j Secondary 22015 43A07 43A10
Key words and phrases Spectrum synthesizable ideals locally compact groups amenability Fourier algebra Figa-Talamanca-Herz algebra
This research was supported by an NSERC grant
Typeset by AMS-TEX
1
2 ZHIGUO HU
1 Introduction
Let A be a semisimple commutative regular tauberian Banach algebra with spectrum
~A In this paper elements of ~A are considered as multiplicative functionals on A and
~A has the Gelfand topology induced by O(A A) Let I be a proper closed ideal of A
with the zero set Z(I) = F The ideal I is said to be synthesizable if I is the largest
closed ideal of A whose zero set is F This definition was given by De Vito for A = LllR)
(where he said that I can be synthesized) and by Ulger recently for general A (see [8]
and [24] respectively)
De Vito proved in [8] that synthesizable ideals of Ll(lR) are of the form I = a E
LllR) tp a = O for some almost periodic function rp on IR It is well-known that
the algebra of almost periodic functions on IR is identified with span ~Ll (It) To study
synthesizable ideals for general algebras A Ulger [24] defined the norm spectrum O tp)
for elements rp of span~A by Orp) = tp a a E A n~A which coincides with the
definition given for instance by Katznelson [19 p159] in the case where A = Ll(IR)
It is showed that Otp) =f 0 for all tp E Span~Ll(R) O (see Katznelsons book [19
Uniqueness theorem p1631) An algebra A with this property is said to have property
(8) Among many other results on the space ~A under the assumption that A has
property (S) plus the separating ball property (SBP for short see sect2 in the sequel)
Ulger gave the following generalization of De Vitos result the ideal I is synthesizable
with a separable zero set iff I = I for some tp E span~A O (see [24 Theorem 551)
It would therefore be important to decide when the algebra A has property (8) In
particular it is interesting to consider this problem for the Fourier algebra A(G) and the
3 SPECTRUM AND AMENABILITY
Figa-Talamanca-Herz algebra A(G) of a locally compact group G We will provide in
this paper an answer to the above question
Let M~(G) donote the C-algebra generated by left translation operators on L2G)
and Gd denote the group G considered as a discrete group Then M~~) is the reduced
C-algebra of Gd A natural question is when M~G) M~~) (which is also denoted
by CG) C(Gd)) Obviously M~G) = M~(~) if G is discrete Zeller-Meier
proved in (25] that M~G) M~(~) whenever Gd is amenable see also (2 3 9])
Recently Bedos complemented Zeller-Meiers result by showing that Gd is amenable iff
G is amenable and M~G) M~~) (l Theorem 3]) Another problem tackled in
this paper is the existence of any characterization for M~(G) M~~) to hold We
will relate this problem to the property (S) of A(G) Here are some details on the
organization of the paper
sect2 consists of some preliminaries and notations used throughout this paper
We investigate in sect3 some basic properties of the norm spectrum and present some
applications Assume the algebra A has the SBP The following are obtained (l) The
space EA is discrete iff 11 rp) = ZI~) for all rp E span EA (Theorem 34) (2) The ideal I is
synthesizable with a separable zero set iff I = I~ for some 11 rp) E span EA O satisfying
the condition tp a=O implies 11rp a) = 0 (Theorem 310) (3) If the algebra AG)
has property (S) then either MG) = PFG) or MG)nPFG) = to (Corollary
314(f)) where MG) and PFG) denote the norm closures of 11G) and LlG) in
AG) respectively The proofs are primarily motivated by some results of Ulger [24]
and our understanding of the relation between the norm spectrum and invariant means
4 ZHIGUO HU
sect4 concerns itself with the property (S) for A = Ap(G) Let G be a locally compact
group with unit e Let tr be the finite trace on the C-algebra M~(Cl) defined by
tr(tp) = tp( e) (tp E [I ( Graquo We prove that A(G) has property (S) if and only if tr is
faithful on Mg((l) if and only if M~(G) ~ M~(~) (Theorem 45) As we know when
G is abelin M~(G) AP(G) (the algebra of almost periodic functions on G) and tr
is always faithful on AP(G) Therefore Theorem 43 shows that Ll (G) A( G) has
property (S) for all abelian groups G This generlizes the above mentioned uniqueness
theorem on LI(R) We further prove that Gd is amenable if and only if G is amenable
and A(G) has property (S) if and only if G is amenable and Ap(G) has property (S) for
some (and hence all) p Our approach depends heavily on the well-developped theory of
Fourier algebra and amenability
This paper is mainly inspired by Ulger [24] The author would like to express her
gratitude to Professor Ali Ulger for his encouragement and valuable suggestions and for
providing his paper [24] and drawing our attention to reference [1]
2 Preliminaries and Some Notations
In this paper we assume that all spaces are over the complex field C For a Banach
space E E and El denote the Banach dual of E and the closed unit ball of E respecshy
tively If tp E E and x E E the value of rp at x will be written as (tp x) or (x rp) We
always regard E as being naturally embedded into its second dual E
Let A be a semisimple commutative regular tauberian Banach algebra with the specshy
trum ~A We consider each element of ~A as a multiplicative functional on A The usual
5 SPECTRUM AND AMENABILITY
(Gelfand) topology of EA is the relative weak topology on EA induced by o(A A)
span EA denotes the norm closed linear subspace of A spanned by EA For a E A and
f E A f a E A is defined by (J a b) = (J ab) b E A If f E A and the set
f a a E Ad is relatively compact f is said to be almost periodic For all P E EA
and a E A P a = (P a) P So each P E EA is almost periodic
For a closed ideal I of A ZI) denotes the zero set of I that is Z(I) = f E EA
I ~ ker f A proper closed ideal I of A is said to be JyntheJizable if I = nEZ(I) ker f
(see De Vito [8] for the case A = Ll(R) and Ulger [24] for general A) In other words
if F = Z(I) then I is synthesizable iff I is the largest closed ideal of A whose zero set
is F Note that if Z(I) is an set of spectral synthesis in the usual sense (that is there
is a unique closed ideal of A with zero set equal to Z(I) see for example Hewitt and
Ross [18 sect39]) then I synthesizable the converse is not true (see Remark 38(i) in the
sequel) It is well-known that enen in L1(R) not every closed ideal is synthesizable
(Malliavins theorem) De Vito [8] proved that synthesizable ideals of Ll(R) are of the
form Irp = a E Ll(R) P a = O for some nonzero almost periodic function P on R
ie P E SpanEL1(R) OJ To study synthesizable ideals for general algebras Ulger [24]
defined the norm spectrum 0-(P) for P E span EAI which coincides with the definition
given for instance by Katznelson [19] in the case where A =L1(R)
Definition 21 ([24]) Let P E span EA The norm spectrum ofP is deflned by
Note that 0-(P) is different from the usual w -spectrum of P which is always
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
SPECTRUM OF COMMUTATIVE BANACH ALGEBRAS
AND AMENABLE LOCALLY COMPACT GROUPS
ZHIGUO Hu Department of Mathematics and Statistics
University of Windsor Windsor Ontario Canada N9B 3P4
Abstract Let A be a semisimple commutative regular tauberian Banach algebra
with spectrum EA We study the norm spectra of spanEA and present some applishy
cations to the Fourier algebra A(G) and the Figa-Talamanca-Herz algebra Ap( G)
of any locally compact group G The algebra A is said to have property (5) if
for all ltp E spanEA OJ ltp has a nonempty norm spectrum Let M~(G) denote
the Cmiddot-algebra generated by left translation operators on L2(G) and Gd denote
the discrete group G We prove that A(G) has property (5) iff the canonical trace
on M~(G) is faithful iff M~(G) ~ M~(~) In particular it is concluded that
Ll(G) has property (5) for all abelian groups G which generalizes the uniqueness
theorem on Ll(R) We further prove that Gd is amenable iff G is amenable and
Ap(G) has property (5) for some (and hence all) p
1991 Mathematics Subject Classification Primary 46J20 46J99 43A30 22D25j Secondary 22015 43A07 43A10
Key words and phrases Spectrum synthesizable ideals locally compact groups amenability Fourier algebra Figa-Talamanca-Herz algebra
This research was supported by an NSERC grant
Typeset by AMS-TEX
1
2 ZHIGUO HU
1 Introduction
Let A be a semisimple commutative regular tauberian Banach algebra with spectrum
~A In this paper elements of ~A are considered as multiplicative functionals on A and
~A has the Gelfand topology induced by O(A A) Let I be a proper closed ideal of A
with the zero set Z(I) = F The ideal I is said to be synthesizable if I is the largest
closed ideal of A whose zero set is F This definition was given by De Vito for A = LllR)
(where he said that I can be synthesized) and by Ulger recently for general A (see [8]
and [24] respectively)
De Vito proved in [8] that synthesizable ideals of Ll(lR) are of the form I = a E
LllR) tp a = O for some almost periodic function rp on IR It is well-known that
the algebra of almost periodic functions on IR is identified with span ~Ll (It) To study
synthesizable ideals for general algebras A Ulger [24] defined the norm spectrum O tp)
for elements rp of span~A by Orp) = tp a a E A n~A which coincides with the
definition given for instance by Katznelson [19 p159] in the case where A = Ll(IR)
It is showed that Otp) =f 0 for all tp E Span~Ll(R) O (see Katznelsons book [19
Uniqueness theorem p1631) An algebra A with this property is said to have property
(8) Among many other results on the space ~A under the assumption that A has
property (S) plus the separating ball property (SBP for short see sect2 in the sequel)
Ulger gave the following generalization of De Vitos result the ideal I is synthesizable
with a separable zero set iff I = I for some tp E span~A O (see [24 Theorem 551)
It would therefore be important to decide when the algebra A has property (8) In
particular it is interesting to consider this problem for the Fourier algebra A(G) and the
3 SPECTRUM AND AMENABILITY
Figa-Talamanca-Herz algebra A(G) of a locally compact group G We will provide in
this paper an answer to the above question
Let M~(G) donote the C-algebra generated by left translation operators on L2G)
and Gd denote the group G considered as a discrete group Then M~~) is the reduced
C-algebra of Gd A natural question is when M~G) M~~) (which is also denoted
by CG) C(Gd)) Obviously M~G) = M~(~) if G is discrete Zeller-Meier
proved in (25] that M~G) M~(~) whenever Gd is amenable see also (2 3 9])
Recently Bedos complemented Zeller-Meiers result by showing that Gd is amenable iff
G is amenable and M~G) M~~) (l Theorem 3]) Another problem tackled in
this paper is the existence of any characterization for M~(G) M~~) to hold We
will relate this problem to the property (S) of A(G) Here are some details on the
organization of the paper
sect2 consists of some preliminaries and notations used throughout this paper
We investigate in sect3 some basic properties of the norm spectrum and present some
applications Assume the algebra A has the SBP The following are obtained (l) The
space EA is discrete iff 11 rp) = ZI~) for all rp E span EA (Theorem 34) (2) The ideal I is
synthesizable with a separable zero set iff I = I~ for some 11 rp) E span EA O satisfying
the condition tp a=O implies 11rp a) = 0 (Theorem 310) (3) If the algebra AG)
has property (S) then either MG) = PFG) or MG)nPFG) = to (Corollary
314(f)) where MG) and PFG) denote the norm closures of 11G) and LlG) in
AG) respectively The proofs are primarily motivated by some results of Ulger [24]
and our understanding of the relation between the norm spectrum and invariant means
4 ZHIGUO HU
sect4 concerns itself with the property (S) for A = Ap(G) Let G be a locally compact
group with unit e Let tr be the finite trace on the C-algebra M~(Cl) defined by
tr(tp) = tp( e) (tp E [I ( Graquo We prove that A(G) has property (S) if and only if tr is
faithful on Mg((l) if and only if M~(G) ~ M~(~) (Theorem 45) As we know when
G is abelin M~(G) AP(G) (the algebra of almost periodic functions on G) and tr
is always faithful on AP(G) Therefore Theorem 43 shows that Ll (G) A( G) has
property (S) for all abelian groups G This generlizes the above mentioned uniqueness
theorem on LI(R) We further prove that Gd is amenable if and only if G is amenable
and A(G) has property (S) if and only if G is amenable and Ap(G) has property (S) for
some (and hence all) p Our approach depends heavily on the well-developped theory of
Fourier algebra and amenability
This paper is mainly inspired by Ulger [24] The author would like to express her
gratitude to Professor Ali Ulger for his encouragement and valuable suggestions and for
providing his paper [24] and drawing our attention to reference [1]
2 Preliminaries and Some Notations
In this paper we assume that all spaces are over the complex field C For a Banach
space E E and El denote the Banach dual of E and the closed unit ball of E respecshy
tively If tp E E and x E E the value of rp at x will be written as (tp x) or (x rp) We
always regard E as being naturally embedded into its second dual E
Let A be a semisimple commutative regular tauberian Banach algebra with the specshy
trum ~A We consider each element of ~A as a multiplicative functional on A The usual
5 SPECTRUM AND AMENABILITY
(Gelfand) topology of EA is the relative weak topology on EA induced by o(A A)
span EA denotes the norm closed linear subspace of A spanned by EA For a E A and
f E A f a E A is defined by (J a b) = (J ab) b E A If f E A and the set
f a a E Ad is relatively compact f is said to be almost periodic For all P E EA
and a E A P a = (P a) P So each P E EA is almost periodic
For a closed ideal I of A ZI) denotes the zero set of I that is Z(I) = f E EA
I ~ ker f A proper closed ideal I of A is said to be JyntheJizable if I = nEZ(I) ker f
(see De Vito [8] for the case A = Ll(R) and Ulger [24] for general A) In other words
if F = Z(I) then I is synthesizable iff I is the largest closed ideal of A whose zero set
is F Note that if Z(I) is an set of spectral synthesis in the usual sense (that is there
is a unique closed ideal of A with zero set equal to Z(I) see for example Hewitt and
Ross [18 sect39]) then I synthesizable the converse is not true (see Remark 38(i) in the
sequel) It is well-known that enen in L1(R) not every closed ideal is synthesizable
(Malliavins theorem) De Vito [8] proved that synthesizable ideals of Ll(R) are of the
form Irp = a E Ll(R) P a = O for some nonzero almost periodic function P on R
ie P E SpanEL1(R) OJ To study synthesizable ideals for general algebras Ulger [24]
defined the norm spectrum 0-(P) for P E span EAI which coincides with the definition
given for instance by Katznelson [19] in the case where A =L1(R)
Definition 21 ([24]) Let P E span EA The norm spectrum ofP is deflned by
Note that 0-(P) is different from the usual w -spectrum of P which is always
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
2 ZHIGUO HU
1 Introduction
Let A be a semisimple commutative regular tauberian Banach algebra with spectrum
~A In this paper elements of ~A are considered as multiplicative functionals on A and
~A has the Gelfand topology induced by O(A A) Let I be a proper closed ideal of A
with the zero set Z(I) = F The ideal I is said to be synthesizable if I is the largest
closed ideal of A whose zero set is F This definition was given by De Vito for A = LllR)
(where he said that I can be synthesized) and by Ulger recently for general A (see [8]
and [24] respectively)
De Vito proved in [8] that synthesizable ideals of Ll(lR) are of the form I = a E
LllR) tp a = O for some almost periodic function rp on IR It is well-known that
the algebra of almost periodic functions on IR is identified with span ~Ll (It) To study
synthesizable ideals for general algebras A Ulger [24] defined the norm spectrum O tp)
for elements rp of span~A by Orp) = tp a a E A n~A which coincides with the
definition given for instance by Katznelson [19 p159] in the case where A = Ll(IR)
It is showed that Otp) =f 0 for all tp E Span~Ll(R) O (see Katznelsons book [19
Uniqueness theorem p1631) An algebra A with this property is said to have property
(8) Among many other results on the space ~A under the assumption that A has
property (S) plus the separating ball property (SBP for short see sect2 in the sequel)
Ulger gave the following generalization of De Vitos result the ideal I is synthesizable
with a separable zero set iff I = I for some tp E span~A O (see [24 Theorem 551)
It would therefore be important to decide when the algebra A has property (8) In
particular it is interesting to consider this problem for the Fourier algebra A(G) and the
3 SPECTRUM AND AMENABILITY
Figa-Talamanca-Herz algebra A(G) of a locally compact group G We will provide in
this paper an answer to the above question
Let M~(G) donote the C-algebra generated by left translation operators on L2G)
and Gd denote the group G considered as a discrete group Then M~~) is the reduced
C-algebra of Gd A natural question is when M~G) M~~) (which is also denoted
by CG) C(Gd)) Obviously M~G) = M~(~) if G is discrete Zeller-Meier
proved in (25] that M~G) M~(~) whenever Gd is amenable see also (2 3 9])
Recently Bedos complemented Zeller-Meiers result by showing that Gd is amenable iff
G is amenable and M~G) M~~) (l Theorem 3]) Another problem tackled in
this paper is the existence of any characterization for M~(G) M~~) to hold We
will relate this problem to the property (S) of A(G) Here are some details on the
organization of the paper
sect2 consists of some preliminaries and notations used throughout this paper
We investigate in sect3 some basic properties of the norm spectrum and present some
applications Assume the algebra A has the SBP The following are obtained (l) The
space EA is discrete iff 11 rp) = ZI~) for all rp E span EA (Theorem 34) (2) The ideal I is
synthesizable with a separable zero set iff I = I~ for some 11 rp) E span EA O satisfying
the condition tp a=O implies 11rp a) = 0 (Theorem 310) (3) If the algebra AG)
has property (S) then either MG) = PFG) or MG)nPFG) = to (Corollary
314(f)) where MG) and PFG) denote the norm closures of 11G) and LlG) in
AG) respectively The proofs are primarily motivated by some results of Ulger [24]
and our understanding of the relation between the norm spectrum and invariant means
4 ZHIGUO HU
sect4 concerns itself with the property (S) for A = Ap(G) Let G be a locally compact
group with unit e Let tr be the finite trace on the C-algebra M~(Cl) defined by
tr(tp) = tp( e) (tp E [I ( Graquo We prove that A(G) has property (S) if and only if tr is
faithful on Mg((l) if and only if M~(G) ~ M~(~) (Theorem 45) As we know when
G is abelin M~(G) AP(G) (the algebra of almost periodic functions on G) and tr
is always faithful on AP(G) Therefore Theorem 43 shows that Ll (G) A( G) has
property (S) for all abelian groups G This generlizes the above mentioned uniqueness
theorem on LI(R) We further prove that Gd is amenable if and only if G is amenable
and A(G) has property (S) if and only if G is amenable and Ap(G) has property (S) for
some (and hence all) p Our approach depends heavily on the well-developped theory of
Fourier algebra and amenability
This paper is mainly inspired by Ulger [24] The author would like to express her
gratitude to Professor Ali Ulger for his encouragement and valuable suggestions and for
providing his paper [24] and drawing our attention to reference [1]
2 Preliminaries and Some Notations
In this paper we assume that all spaces are over the complex field C For a Banach
space E E and El denote the Banach dual of E and the closed unit ball of E respecshy
tively If tp E E and x E E the value of rp at x will be written as (tp x) or (x rp) We
always regard E as being naturally embedded into its second dual E
Let A be a semisimple commutative regular tauberian Banach algebra with the specshy
trum ~A We consider each element of ~A as a multiplicative functional on A The usual
5 SPECTRUM AND AMENABILITY
(Gelfand) topology of EA is the relative weak topology on EA induced by o(A A)
span EA denotes the norm closed linear subspace of A spanned by EA For a E A and
f E A f a E A is defined by (J a b) = (J ab) b E A If f E A and the set
f a a E Ad is relatively compact f is said to be almost periodic For all P E EA
and a E A P a = (P a) P So each P E EA is almost periodic
For a closed ideal I of A ZI) denotes the zero set of I that is Z(I) = f E EA
I ~ ker f A proper closed ideal I of A is said to be JyntheJizable if I = nEZ(I) ker f
(see De Vito [8] for the case A = Ll(R) and Ulger [24] for general A) In other words
if F = Z(I) then I is synthesizable iff I is the largest closed ideal of A whose zero set
is F Note that if Z(I) is an set of spectral synthesis in the usual sense (that is there
is a unique closed ideal of A with zero set equal to Z(I) see for example Hewitt and
Ross [18 sect39]) then I synthesizable the converse is not true (see Remark 38(i) in the
sequel) It is well-known that enen in L1(R) not every closed ideal is synthesizable
(Malliavins theorem) De Vito [8] proved that synthesizable ideals of Ll(R) are of the
form Irp = a E Ll(R) P a = O for some nonzero almost periodic function P on R
ie P E SpanEL1(R) OJ To study synthesizable ideals for general algebras Ulger [24]
defined the norm spectrum 0-(P) for P E span EAI which coincides with the definition
given for instance by Katznelson [19] in the case where A =L1(R)
Definition 21 ([24]) Let P E span EA The norm spectrum ofP is deflned by
Note that 0-(P) is different from the usual w -spectrum of P which is always
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
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being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
3 SPECTRUM AND AMENABILITY
Figa-Talamanca-Herz algebra A(G) of a locally compact group G We will provide in
this paper an answer to the above question
Let M~(G) donote the C-algebra generated by left translation operators on L2G)
and Gd denote the group G considered as a discrete group Then M~~) is the reduced
C-algebra of Gd A natural question is when M~G) M~~) (which is also denoted
by CG) C(Gd)) Obviously M~G) = M~(~) if G is discrete Zeller-Meier
proved in (25] that M~G) M~(~) whenever Gd is amenable see also (2 3 9])
Recently Bedos complemented Zeller-Meiers result by showing that Gd is amenable iff
G is amenable and M~G) M~~) (l Theorem 3]) Another problem tackled in
this paper is the existence of any characterization for M~(G) M~~) to hold We
will relate this problem to the property (S) of A(G) Here are some details on the
organization of the paper
sect2 consists of some preliminaries and notations used throughout this paper
We investigate in sect3 some basic properties of the norm spectrum and present some
applications Assume the algebra A has the SBP The following are obtained (l) The
space EA is discrete iff 11 rp) = ZI~) for all rp E span EA (Theorem 34) (2) The ideal I is
synthesizable with a separable zero set iff I = I~ for some 11 rp) E span EA O satisfying
the condition tp a=O implies 11rp a) = 0 (Theorem 310) (3) If the algebra AG)
has property (S) then either MG) = PFG) or MG)nPFG) = to (Corollary
314(f)) where MG) and PFG) denote the norm closures of 11G) and LlG) in
AG) respectively The proofs are primarily motivated by some results of Ulger [24]
and our understanding of the relation between the norm spectrum and invariant means
4 ZHIGUO HU
sect4 concerns itself with the property (S) for A = Ap(G) Let G be a locally compact
group with unit e Let tr be the finite trace on the C-algebra M~(Cl) defined by
tr(tp) = tp( e) (tp E [I ( Graquo We prove that A(G) has property (S) if and only if tr is
faithful on Mg((l) if and only if M~(G) ~ M~(~) (Theorem 45) As we know when
G is abelin M~(G) AP(G) (the algebra of almost periodic functions on G) and tr
is always faithful on AP(G) Therefore Theorem 43 shows that Ll (G) A( G) has
property (S) for all abelian groups G This generlizes the above mentioned uniqueness
theorem on LI(R) We further prove that Gd is amenable if and only if G is amenable
and A(G) has property (S) if and only if G is amenable and Ap(G) has property (S) for
some (and hence all) p Our approach depends heavily on the well-developped theory of
Fourier algebra and amenability
This paper is mainly inspired by Ulger [24] The author would like to express her
gratitude to Professor Ali Ulger for his encouragement and valuable suggestions and for
providing his paper [24] and drawing our attention to reference [1]
2 Preliminaries and Some Notations
In this paper we assume that all spaces are over the complex field C For a Banach
space E E and El denote the Banach dual of E and the closed unit ball of E respecshy
tively If tp E E and x E E the value of rp at x will be written as (tp x) or (x rp) We
always regard E as being naturally embedded into its second dual E
Let A be a semisimple commutative regular tauberian Banach algebra with the specshy
trum ~A We consider each element of ~A as a multiplicative functional on A The usual
5 SPECTRUM AND AMENABILITY
(Gelfand) topology of EA is the relative weak topology on EA induced by o(A A)
span EA denotes the norm closed linear subspace of A spanned by EA For a E A and
f E A f a E A is defined by (J a b) = (J ab) b E A If f E A and the set
f a a E Ad is relatively compact f is said to be almost periodic For all P E EA
and a E A P a = (P a) P So each P E EA is almost periodic
For a closed ideal I of A ZI) denotes the zero set of I that is Z(I) = f E EA
I ~ ker f A proper closed ideal I of A is said to be JyntheJizable if I = nEZ(I) ker f
(see De Vito [8] for the case A = Ll(R) and Ulger [24] for general A) In other words
if F = Z(I) then I is synthesizable iff I is the largest closed ideal of A whose zero set
is F Note that if Z(I) is an set of spectral synthesis in the usual sense (that is there
is a unique closed ideal of A with zero set equal to Z(I) see for example Hewitt and
Ross [18 sect39]) then I synthesizable the converse is not true (see Remark 38(i) in the
sequel) It is well-known that enen in L1(R) not every closed ideal is synthesizable
(Malliavins theorem) De Vito [8] proved that synthesizable ideals of Ll(R) are of the
form Irp = a E Ll(R) P a = O for some nonzero almost periodic function P on R
ie P E SpanEL1(R) OJ To study synthesizable ideals for general algebras Ulger [24]
defined the norm spectrum 0-(P) for P E span EAI which coincides with the definition
given for instance by Katznelson [19] in the case where A =L1(R)
Definition 21 ([24]) Let P E span EA The norm spectrum ofP is deflned by
Note that 0-(P) is different from the usual w -spectrum of P which is always
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
4 ZHIGUO HU
sect4 concerns itself with the property (S) for A = Ap(G) Let G be a locally compact
group with unit e Let tr be the finite trace on the C-algebra M~(Cl) defined by
tr(tp) = tp( e) (tp E [I ( Graquo We prove that A(G) has property (S) if and only if tr is
faithful on Mg((l) if and only if M~(G) ~ M~(~) (Theorem 45) As we know when
G is abelin M~(G) AP(G) (the algebra of almost periodic functions on G) and tr
is always faithful on AP(G) Therefore Theorem 43 shows that Ll (G) A( G) has
property (S) for all abelian groups G This generlizes the above mentioned uniqueness
theorem on LI(R) We further prove that Gd is amenable if and only if G is amenable
and A(G) has property (S) if and only if G is amenable and Ap(G) has property (S) for
some (and hence all) p Our approach depends heavily on the well-developped theory of
Fourier algebra and amenability
This paper is mainly inspired by Ulger [24] The author would like to express her
gratitude to Professor Ali Ulger for his encouragement and valuable suggestions and for
providing his paper [24] and drawing our attention to reference [1]
2 Preliminaries and Some Notations
In this paper we assume that all spaces are over the complex field C For a Banach
space E E and El denote the Banach dual of E and the closed unit ball of E respecshy
tively If tp E E and x E E the value of rp at x will be written as (tp x) or (x rp) We
always regard E as being naturally embedded into its second dual E
Let A be a semisimple commutative regular tauberian Banach algebra with the specshy
trum ~A We consider each element of ~A as a multiplicative functional on A The usual
5 SPECTRUM AND AMENABILITY
(Gelfand) topology of EA is the relative weak topology on EA induced by o(A A)
span EA denotes the norm closed linear subspace of A spanned by EA For a E A and
f E A f a E A is defined by (J a b) = (J ab) b E A If f E A and the set
f a a E Ad is relatively compact f is said to be almost periodic For all P E EA
and a E A P a = (P a) P So each P E EA is almost periodic
For a closed ideal I of A ZI) denotes the zero set of I that is Z(I) = f E EA
I ~ ker f A proper closed ideal I of A is said to be JyntheJizable if I = nEZ(I) ker f
(see De Vito [8] for the case A = Ll(R) and Ulger [24] for general A) In other words
if F = Z(I) then I is synthesizable iff I is the largest closed ideal of A whose zero set
is F Note that if Z(I) is an set of spectral synthesis in the usual sense (that is there
is a unique closed ideal of A with zero set equal to Z(I) see for example Hewitt and
Ross [18 sect39]) then I synthesizable the converse is not true (see Remark 38(i) in the
sequel) It is well-known that enen in L1(R) not every closed ideal is synthesizable
(Malliavins theorem) De Vito [8] proved that synthesizable ideals of Ll(R) are of the
form Irp = a E Ll(R) P a = O for some nonzero almost periodic function P on R
ie P E SpanEL1(R) OJ To study synthesizable ideals for general algebras Ulger [24]
defined the norm spectrum 0-(P) for P E span EAI which coincides with the definition
given for instance by Katznelson [19] in the case where A =L1(R)
Definition 21 ([24]) Let P E span EA The norm spectrum ofP is deflned by
Note that 0-(P) is different from the usual w -spectrum of P which is always
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
5 SPECTRUM AND AMENABILITY
(Gelfand) topology of EA is the relative weak topology on EA induced by o(A A)
span EA denotes the norm closed linear subspace of A spanned by EA For a E A and
f E A f a E A is defined by (J a b) = (J ab) b E A If f E A and the set
f a a E Ad is relatively compact f is said to be almost periodic For all P E EA
and a E A P a = (P a) P So each P E EA is almost periodic
For a closed ideal I of A ZI) denotes the zero set of I that is Z(I) = f E EA
I ~ ker f A proper closed ideal I of A is said to be JyntheJizable if I = nEZ(I) ker f
(see De Vito [8] for the case A = Ll(R) and Ulger [24] for general A) In other words
if F = Z(I) then I is synthesizable iff I is the largest closed ideal of A whose zero set
is F Note that if Z(I) is an set of spectral synthesis in the usual sense (that is there
is a unique closed ideal of A with zero set equal to Z(I) see for example Hewitt and
Ross [18 sect39]) then I synthesizable the converse is not true (see Remark 38(i) in the
sequel) It is well-known that enen in L1(R) not every closed ideal is synthesizable
(Malliavins theorem) De Vito [8] proved that synthesizable ideals of Ll(R) are of the
form Irp = a E Ll(R) P a = O for some nonzero almost periodic function P on R
ie P E SpanEL1(R) OJ To study synthesizable ideals for general algebras Ulger [24]
defined the norm spectrum 0-(P) for P E span EAI which coincides with the definition
given for instance by Katznelson [19] in the case where A =L1(R)
Definition 21 ([24]) Let P E span EA The norm spectrum ofP is deflned by
Note that 0-(P) is different from the usual w -spectrum of P which is always
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
6 ZHIGUO HU
nonempty if cp is nonzero (see for example Hewitt and Ross [18 sect40]) As mentioned
in the introduction u(cp) -f 0 for all cp E spanEA OJ when A = Ll(R) But it is not
the case for general algebras A Therefore we would like to give the following
Definition 22 The algebra A is said to have property (S) if u(cp) -f 0 for all cp E
span EA OJ
In [24] Ulger introduced the separating ball property (SBP for short) that plays an
important role in his discussion on the discreteness of EA under the weak topology of
Amiddot The algebra A is said to have the SBP if given any two distinct elements I and 9
in EA there exists a E Al such that (I a) = 1 and g a = O For easy reference we
would like to quote the following results from Ulger [24]
Lemma 23 ([24 Lemma 51]) Assume A has the SBP Then for each I E EA there
exists mf E A such that (mfl J) = 1 and (mj g) = 0 for all 9 E EA I
Lemma 24 ([24 Lemma 52 and 53]) Assume A has the SBP Let cp E spanEA OJ
I E EA and a E A Then
(i) (cp a mf) = (I a) (cp mf)
(ii) IE u(cp) iff(cp mf) -f O
(iii) u(cpmiddot a) = u(cp)ng E EA (g a) -f OJ
(iv) u(cp) is a countable subset of EA
Throughout this paper G denotes a locally compact group with unit e and a fixed
left Haar measure For any subset U of G lu denotes the characteristic function of
U LP(G) (1 $ p $ 00) has the usual meaning The group G is said to be amenable if
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
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Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
7 SPECTRUM AND AMENABILITY
there exists m E LOO(G) such that IImll = (m IG) = 1 and (m xl) = (m I) for all
f E LOO(G) and x E G where rJ is the left translate of f by X Gd denotes the algebraic
group G endowed with the discrete topology Then G is amenable if G d is amenable
All solvable groups and all compact groups are known to be amenable However the
free group on two generators is not amenable For more information on this subject see
Greenleafs book [15] and the recent books of Pier [22] and Paterson [21]
For 1 lt p lt 00 Ap(G) denotes the Figa-Talamanca-Herz algebra of G Elements of
Ap(G) can be represented nonuniquely as
() a = l00
Vn un
n=l
with Un E LP(G) Vn E Lq(G) (~ + = 1) un(x) =un(x- 1) and L~=l lIunlip IIvnll q lt
00 The norm of a is defined by
00
lIall = infl lIunlip IIvnll q
n=l
where the infimum is taken over all the possible representations of a as in () It is known
that Ap( G) is a subspace of Co(G) (the space of all continuous functions on G vanishing
at infinity) and equipped with the above norm and the pointwise multiplication is a
semisimple commutative regular tauberian Banach algebra whose spectrum is G (via
Dirac measures) For p = 2 Ap(G) = A( G) the Fourier algebra of Gj for commutative
G with dual group 0 A(G) is isometrically isomorphic to Ll(O) See Eymard [11] and
Herz [17] for details on the algebras A(G) and Ap(G) respectively Furthermore for any
1 lt p lt 00 Ap(G) has the SBP (see Ulger [24 Proposition 25])
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
8 ZHIGUO HU
Let M(G) denote the measure algebra of G and Md(G) the space of discrete measures
in M(G) Then M(G) can be considered as a subspace of Ap(G) by
(p u) = fa u(x)dp(x)
with 1IpIIA(G)- $ 1IpIIM(G) In particular (ox u) = u(x) x E G u E Ap(G) where Ox
denotes the point measure at x
By definition Mp(G) M(G) and PFp(G) are the norm closures of M(G) Md(G)
and Ll (G) in Ap( G) respectively (see Granirer [12]) APp( G) denotes the space of all
almost periodic functionals on Ap(G) It is known that M(G) ~ APp(G) (see Granirer
[12 Proposition 12]) For p = 2 P F2( G) = C(G) the reduced group C -algebra
of G and Mg(G) is also denoted by C(G) (see Lau [20]) Under the identification
G = EA(G) we have spanEA(G) = M(G)
An element m of Ap(G) is said to be a topologically invariant mean on Ap(G)
if IImll = (moe) = 1 and (m Tmiddot u) = (m T) for all T E Ap(G) u E Ap(G) with
II uII = u(e) = 1 Let T I Mp( G) be the set of all topologically invariant means on Ap( G)
It is known that TIMp(G) f 0 (see Renaud [23 p287] for p = 2 and Granirer [12
Theorem 5] for general p)
We know that there are groups G such that Ap( G) fails to have property (S) (see sect4
for details) For this sake we would like to give the following
Definition 25 For 1 lt p lt 00 the group G is said to have property (Sp) if Ap(G) bas
property (5) tbat is u(ltp) f 0 for all P E M(G) o
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
9 SPECTRUM AND AMENABILITY
3 Norm Spectrum Discreteness of 2A and Synthesizable Ideals
Throughout this section A will be a semisimple commutative regular tauberian Bashy
nach algebra and EA be the spectrum of A with the Gelfand topology Then for any
proper closed ideal 1 of A the zero set Z(I) of 1 is nonempty We also assume in this
section that A has the SBP
For f E 2A let mf E A be the same as in Lemma 23 The following lemma is a
direct consequence of Lemma 24(ii)
Lemma 31 (i) For cp = EfEEA cf f E span EA 0() = f E EA cf =i o
(ii) IfltPn = EfEEA cf E span2A andltpn ~ ltp E spanEA then cp mf = limn_ oo c
for all f E EAmiddot In particular O(cp) = f E 2A limn_ooc =i o
11 = a E A P a = O
Then 11 is a closed ideal of A If ltp =i 0 Itp is a proper closed ideal in A
Lemma 32 For any cp E span2A O(cp) ~ Z(Itp)
Proof Let ltp E span2Ao If cp = 0 then O(ltp) = 0 ~ Z(ltp)
Assume that cp =i o Let f E O(cp) We need to show that f E Z(Itp) Let a E Itpo
Then cp bull a = 0 and hence 0 = (ltp a mf) = (j a) (cp mf) (by Lemma 24(i)) But0
ltp mf =i 0 (Lemma 24(ii)) It follows that (j a) = 0 for all a E 11 ie f E Z(I1)
Therefore O(cp) ~ Z(ltp) 0
The following simple lemma is obvious For the sake of completeness we also include
its proof here
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
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[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
10 ZHIGUO HU
Lemma 33 If X is a nondiscrete locally compact Hausdorff space then X contains a
countable non-closed subset
Proof Fix an x EX By the assumption there exists a strictly decreasing sequence
(Un)ngtl of compact neighbourhoods of x For each n take Xn E Un Un+1 Let
D = X n)n21 and y E X be any cluster point ofthe sequence (X n )n2 1 Then y E nn21 Un
and hence y cent D Therefore D is a countable non-closed subset of X 0
We first observe the following relation between the discreteness of EA and the norm
spectra
Theorem 34 The space EA is discrete if and only if for all I() E spanEA ult) =
ZIIf)
Proof Suppose the space EA is discrete Let lt E span EA By Lemma 32 we only
need to show that ZIIf) ~ ult) Let f E ZIIf) Then IIf ~ ker I that is for all a E A
I() bull a = 0 implies (j a) = O Since A is regular and EA is discrete there exists a E A
such that (j a) = 1 and (g a) = 0 for all 9 E EA fl So we can now take mf = a
emf is the same as in Lemma 23) Since I() bull a =F 0 otherwise (I a) = 0) there exists
b E A such that (lt bull a b) =F 0 that is
o=F (I() bull a b = (ltp b mf) = (j b) (lt mf) (by Lemma 24(iraquo
Hence (lt mf) =F O f E ultp) follows readily from Lemma 24ii) Therefore Z(IIf) ~
ult)
Conversely suppose ult) = ZIIf) for all lt E span EA Assume that the space EA is
not discrete By Lemma 33 EA contains a countable non-closed subset (fn)n21 Let
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
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[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
11 SPECTRUM AND AMENABILITY
Cfin = E~=I 21 nIn E span EA By Lemma 31 u(ltp) = (fn)nh which is not closed But
the zero set Z(I) is always closed in the space EA So u(ltp) i= Z(1) a contradiction
Therefore the space E A is discrete 0
As an immediate consequence we have
Corollary 35 If the space EA is discrete then the algebra A has property (8)
Proof Let ltp E span EA OJ Since ltp i= 0 and A is regular tauberian I is a proper
closed ideal of A Thus Z(1) i= 0 By Theorem 34 u(ltp) = Z(I) i= 0 Therefore A
has property (S) 0
Remark 36 The converse of Corollary 35 is not true For example A = A(JR)
Ll(JR) has property (S) (see Katznelsons book (19 p163]) but EA = R is not discrete
In next section we will present a characterization for A(G) to have property (S) for all
locally compact groups G
Next by using norm spectra of elements in spanEA we will investigate the structure
of synthesizable ideals of A For Cfi E spanEA OJ we consider the following conditions
on Cfi
(1) Cfi = E~=I cnln for some Cn E C and In E spanEA with (u(fn))n1 prurwIse
disjoint
(2) For all a E A ltp a i= 0 implies that u(ltp a) i= 0
(3) u(ltp) is (weak) dense in Z(1)
(4) u(ltp) i= 0
(5) The ideal I is synthesizable
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
12 ZHIGUO HU
We have the following implications
Proposition 37 Let ltp E spanEA OJ Then (1) (2) (3) (4) and (2) (5)
Proof (1) (2) follows from Lemma 31 and (3) (4) is trivial
We now follow an argument of Ulger [24 Theorem 55] to prove (2) (3) and (2)
(5)
(2) =gt (3) Suppose (2) holds for ltpo Assume that O(ltp) is not dense in Z(Iltp) Then
there exists fEZ(Iltp) such that f is not in the (weakmiddot) closure of 0(tp ) By the
regularity of A there is a E A such that (I a) j 0 and (g a) = 0 for all 9 E 0(ltp) Thus
tp a j O By the assumption of condition (2) O(ltp a) j 0 But by Lemma 24(iii)
O(ltpmiddot a) = O(ltp)ng E EA (g a)j OJ SO there exists 9 E O(tp) such that (g a)j 0
a contradiction Therefore 0(ltp) is dense in Z(Iltp)
(2) =gt (5) Suppose tp satisfies condition (2) Let J = Iltp We need to prove that
J = nEZ(J) ker fmiddot Clearly J ~ nEZ(J) ker f To prove nEZ(J) ker f ~ J let a E
nEZ(J) ker fmiddot Then (I a) = 0 for all f E Z(J)
We claim that ltp a = O Otherwise by condition (2) O(tp a) = g E EA (g a) j
O j 0 Thus there is 9 E O(ltp) such that (g a) j O However by Lemma 32
O(ltp) ~ Z(Iltp) We have (g a) = 0 a contradiction Hence tpmiddota = 0 that is a E lltp = J
Therefore J = nEZ(J) ker f It follows that Iltp is synthesizable 0
Remark 38 (i) Let E be a closed subset of EA Denote I(E) = nEE ker f Then
I(E) is the largest closed ideal of A whose zero set is E The set E is said to be an set
of spectral synthesis (s-set for short) if I(E) is the only closed ideal of A with zero set
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
13 SPECTRUM AND AMENABILITY
E (cf Hewitt and Ross [18 sect39]) Let I be a proper closed ideal of A By definition
if Z(I) is an s-set then I = I(Z(I)) = nEZ(I) ker f is synthesizable The converse is
not true even for A = A(JR) and ideals of the form IV It is well-known that R has a
closed subset E which is not an s-set for A(JR) (Malliavins theorem) Suppose (X n )ngtl
1 is a dense subset of E and let rp = l=1 2n 6xn E span EA(lIt) By Proposition 37 IV is
synthesizable but Z (IV) = 0(rp) = E is not an s-set On the other hand it is true that
each proper closed ideal of A is synthesizable iff each closed subset of EA is an s-set
(ii) Let G be an infinite compact group with unit e and the normalized Haar measure A
Chou Lau and Rosenblatt [5] called G having property (A) if AE M~(n = span EA(G)
Suppose G has property (A) (for example G = SO(n) n ~ 3 see Chou Lau and
Rosenblatt [5 p343] and Chou [4 p240D The closed ideal h of A(G) is O and
hence I) is synthesizable But O(A) = 0 (see Lemma 313 in the sequel) Therefore the
synthesizability of IV does not imply that O(rp) =F 0 If in the above we take rp = A+6e
then 0( rp) = e f 0 while 0( rp) is not dense in Z(IV) (= G) If we further assume
1 that G is separable with dense subset (Xn)n21 and let rp = A+ l=1 2n6x then rp
satisfies condition (2) but not condition (1) So we do not have [(4) =gt (3)] or [(2) =gt
(1)] The implication (1) =gt rp E P(EA ) fails either see the example given by Cowling
and Fournier in [6 p64-65] We do not know whether the implication (3) =gt (2) is true
We are only able to show that [(3) and (5)] =gt (2)
However (2) (3) and (4) are equivalent if they hold for all rp E span EA OJ This is
the following corollary which follows readily from Proposition 37 and is in fact included
in the proof of llger [24 Theorem 55]
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
14 ZHIGUO HU
Corollary 39 The following assertions are equivalent
(a) The algebra A has property (8)
(b) For all cp E spanEA O q(cp) is (weakmiddot) dense in Z(I)
(c) For all cp E spanEA O cp satisfies condition (2)
Let J be a proper closed ideal of A Under the assumption that A has property (5)
Ulger [24 Theorem 55] proved that I is synthesizable with (weakmiddot) separable zero set
iff I = IIgt for some cp E span EA O This generalizes De Vitos result on synthesizable
ideals of Ll(R) (see [8]) We observe that only condition (2) was used in Ulgers proof
(not the property (5) on the whole algebra A) Therefore we have the following slightly
strong assertion
Theorem 310 Let I be a proper closed ideal of A Then the following statements are
equivalent
(a) I is synthesizable and Z(I) is (weakmiddot) separable
(b) J = IIgt for some cp E span EA O satisfying condition (2)
Proof (b) (a) It follows from Proposition 37 and Lemma 24(iv)
(a) (b) Assume that the ideal I is synthesizable and Z(I) is weakmiddot separable
Let (fn)n~l be a weakmiddot dense sequence in Z(I) Let cp = E~=l 21n in Then cp E
spanEA OJ and cp satisfies condition (1) (hence condition (2)) Now q(cp) = (fn)n~l
(by Lemma 31) By Proposition 37 IIgt is synthesizable and Z(IIraquo = q(cp)wmiddot = Z(J)
Therefore 1= nEz(I) ker i =nEz(I) ker f = IIgt 0
Corollary 311 ([24]) Assume that the algebra A has property (5) and EA is (weakmiddot)
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
15 SPECTRUM AND AMENABILITY
separable Then a proper closed ideal I of A is synthesizable if and only if I = lrp for
BomeP E span EA a
In the rest of this section we give some applications of the results obtained to the
Figa-Talamanca-Herz algebra Ap(G) Let G be a locally compact group with unit e
and a fixed left Haar measure A Let 1 lt P lt 00 Recall that Ap(G) is a semisimple
commmutative regular tauberian Banach algebra with spectrum G (via Dirac measures)
and span EAp(G) = M(G) Also Ap(G) has the SBP For x E G the set of topologically
invariant means on Ap(G) at x is defined by TlMp(x) = m E Ap(G) IImll =
m or = 1 and (m Tmiddot u) (m T) for all T E Ap(G) u E Ap(G) lIuli = u(x) = I
(see Granirer [13] and [14]) In particular TlMp(e) = TlMp(G) the set of topologically
invariant means on Ap(G) (see sect2) It is well-known that TlMp(G) 1= 0 (see Renaud
[23 p287] for p = 2 and Granirer [12 Theorem 5] for general p) And it is easy
to see that for all x E G TlMp(x) = rm E Ap(G) m E TlMp(G) where
(rm T) = (m r-1T) and (rT u) = (T ru) for all T E Ap(G) and u E Ap(G) (ru
denotes the left translate of u by x)
For A = Ap( G) concerning the functional m I in Lemma 23 we have the following
observation
Lemma 312 Let A = Ap(G) (1 lt p lt 00) and x E G Then for each m E TlMp(G)
rm can be taken as the functional mr as in Lemma 23
Proof Let m E TlMp(G) and x E G Then rm E TlMp(x) Thus rm or = 1 We
only need to show that (rm 0) = 0 for all y E G x To prove this let y E G x
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
16 ZHIGUO HU
1 ~ Choose a compact neighbourhood Uofesuch that xUnyU = 0 Letu = A(U)l x ul u E
Ap(G) Then lIuli =u(x) =1 and u(y) = O Now 8x u =u(y)8y =O Therefore
The proof is complete 0
The following lemma will be useful in the sequel It shows that if I E M(G) nM(G)
then the norm spectrum u(p) of I is closely related to the discrete part of the measure
I and is independent of the number p
Lemma 313 Let A = Ap(G) (1 lt p lt 00) and mE TIMp(G)
(a) If I E M(G) then for all x E G (1 xm) = p(x) In particular if I E
M( G) nM(G) then
u(p) = x E G 1(x) tf O
(b) If G is nondiscrete then for all ltp E PF(G) nM(G) u(ltp) = 0
Proof (a) Let I E M(G) and x E G Then x-II E M(G) is the measure given by
x-lp(E) = p(xE) for all measurable sets E By Granirer [12 Proposition 10]
(1 xm) (m x-lp) = x-1p(e) = p(x)
If I E M(G)nM(G) then x E u(p) iff (11 xm) tf 0 (by Lemma 24(ii) and Lemma
312) iff p(x) tf O The second statement follows
(b) Suppose Gis nondiscrete and ltp E PFp(G) nM(G) Then there exists a sequence
(n)nl in Ll(G) such that n -+ ltp in the 1ImiddotIIAp (G)-norm For all x E G we have
(ltp xm) = lim Un xm =0 (by part (a))n-oo
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
17 SPECTRUM AND AMENABILITY
It follows from Lemma 24(ii) and Lemma 312 that O(P) = 0 0
For any T E Ap(G) the support of T is defined as follows Let x E G Then
x ~ supp T iff there is a neighbourhood U of x such that (T u) = 0 for all u E Ap(G)
with supp u ~ U An equivalent definition for supp T is that x E supp T iff T u = 0
implies u(x) = 0 for all u E Ap(G) (see Herz [17]) LetP E M(G) By definition
11 = u E Ap(G) P u = OJ and hence Z(IP) = supPPmiddot
To conclude this section we would like to present the following corollary as a summary
of the applications to Ap(G) of 32343539311 and 313
Corollary 314 Let G be a locally compact group and A =Ap( G) (1 lt p lt 00) Then
the following assertions hold
(a) For all cp E M(G) O(cp) ~ suppcp
(b) G is discrete jffO(cp) = suppcp for all cp E M(G)
(c) If G is discrete then G has property (Sp)
(d) G has property (Sp) jff O(cp) is dense in sUPPP for all cp E M(G)
(e) Suppose G is second countable and G has property (Sp) Then a proper closed
ideal I of Ap(G) is synthesizable iff 1= 11 for some cp E M(G) OJ
(f) IfG has property (Sp) then either M(G) = PFp(G) or M(G) npFp(G) =
OJ
4 Property (Sp) Faithful trace and Amenability of G
From Corollary 311 we see that it is interesting to consider when an algebra A has
property (S) Here we would like to investigate this question for A = Ap(G)
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
18 ZHIGUO HU
Throughout this section G denotes a locally compact group with unit e and a fixed left
Haar measure A Recall that for 1 lt p lt 00 the Figa-Talamanca-Herz algebra Ap(G)
is a semisimple commutative regular tauberian Banach algebra and has the SBP The
spectrum of Ap(G) is identified with G via Dirac measures and SpanEApG) =M~(G) ~
APp(G) Also the group G has property (Sp) iff u(cp) = 0 for all cp E M~(G) OJ
(sect2) For p = 2 A2(G) = A(G) the Fourier algebra of G and Mg(G) is the C-algebra
generated by left translation operators on L2(G) Also A2(G) = VN(G) the von
Neumann algebra generated by the left regular representation of G See Eymard [11) for
details on the algebras A(G) and V N (G)
Since Mg(G) is a C-algebra let us start with the property (S2)
Let m E TIM2(G) a topologically invariant mean on VN(G) It is known that
(m Jt) = Jt(e) for all Jt E M(G) (see Dunkl and Ramirez [10 Theorem 211 and
Chapter 8]) Let tr = mIMl(G) the restriction of m to M2(G) The functional tr has
the following property
(1) tr(Jtv) =tr(vJL) = LXEGJL(x)v(x- 1 ) Jt v E M(G)
(2) tr(Jt Jt) LxEG IJL( x )12 ~ 0 JL E M(G)
Therefore tr is the unique finite trace on the C-algebra M 2 ( G) with tr(JL) = JL( e)
Jt E M(G) The trace tr is said to be faithful on M~(G) if tr(cpcp) = 0 implies that
cp = 0 for all cp E Mg(G) where ltfI denotes the adjoint of ltfI as a bounded operator on
L2(G)
We first establish the following lemma whose proof constitutes the major technical
part of this paper
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
19 SPECTRUM AND AMENABILITY
Lemma 41 H the trace tr is faithful on Mg(a) then the group G has property (52)
Proof Suppose the trace tr is faithful on M~(a) Let tp E M~(a) o We must
show that 0(tp) I- 0
Let (Uo)o be a basic neighbourhood system of e with each Uo compact For each a
yen
let U o = A(U 1
) IVa lua Then U o E A(G) Iluoll = uo(e) = 1 and o
Thus suppUo UoUo -1 for all a
ci ct E C and xi xkn are distinct elements of G Note that A(G) nCoo(G)
is dense in A(G) (Coo(G) denotes the space of continuous functions on G with compact
support) and 0( tp u) = 0(tp) nx E G u(x) I- o Replacing Pn by tpn bullu and tp by tp u
for some u E A(G) nCoo(G) we may assume that there exists a compact subset K of
G such that supptp K and SUPPPn ~ K for all n Also we may assume that 1Itp1l =1
and lItpnll = 1 for all n For each n choose an index an such that (xiUQnh9Skn is
pairwise disjoint
For T E V N(G) = A(G) and u E A(G) Eymard denoted T E V N(G) and Tu E
A(G) by
(T v) = (T ii) v E A(G)
and
(5 Tu) = (T5 u) 5 E VN(G)
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
20 ZHIGUO HU
where v(x) = v(x-I) HuE A(G) and supp u is compact then Tu = T( u) the evaluation
of the bounded operator Ton L2(G) at u (see Eymard [11 p213])
Now for each n let 9n = PnUa E A(G) Then
Recall that 6z(1) = z-11 for x E G and 1 E L2(G) where zl denotes the left translate
of 1 by x Therefore for all x E G we have
kn
= Lci(zi)-lUan(X) i=1
k
=Lci uan laquoxi)-lx) i=l
_ ~n (xfUa nxUa )
- Lt ci (U) i=1 a
Thus
k k
Pn 9n =L Ci9n(xi)6zr =L Icil26zr i=1 i=l
because (xfUanhltiltk is pairwise disjoint
a convergent subsequence We may assume that Pn 9n ~ T E VN(G) Note that
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
21 SPECTRUM AND AMENABILITY
On the other hand we have
Ion
= limtr( crc~6(z~)-lz~) (ltp = L Ci 6(Zi)-1)n ~ J I
l~ij~kn i=l
= lin L cf cj 6(zgt-lzi (e) 1~ij9n
Ion
= lim L Icil2 bull
n i=l
Note that UltPn gn II M(G) =E~I Ici 12 So (ltPn bullgn )n~1 is a bounded sequence in M(G)
Therefore (ltpn bull gn)ngt1 has a subnet weak -convergent to some I E M(G) = Co(G)
Recall that A(G) ~ Co(G) and for u E A(G)
(p UM(G)Co(G) = u(x)dl(x)L =(I U)VN(G)A(G)
It follows that T = I and hence T E M(G) nM~(G)
By the regularity of A(G) we can take U E A(G) such that U = 1 on the compact set
K Recall that supp ltpn ~ K for all n and note that tr is faithful on M~(G) We have
(T u) = lim (ltPn bull gn u) n
10
=lim(lcrI26z~ u) n LJ bull
i-I
2 = linL
10
Icili-I
= tr (ltPltp) gt O
Therefore T =F 0 that is T E M(G) nM~(G) O
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
22 ZHIGUO HU
Let T = Td + Tc where Td E ~1d(G) and Tc E Mc( G) (the space of continuous
measures in M(G)) We claim that Td =f O Otherwise
tr (TT) = tr (TTc) I
= 0 (because TTc E Mc(G))
Since tr is faithful on M~(G) T = 0 a contradiction Therefore T has nonzero discrete
part Td By Lemma 313(a) u(T) =f 0 Note that cp9n -+ T By definition u(T) ~ u(P)
It follows that u(ltp) =f 0 Since cP E M(G) O is arbitary the group G has property
(52) The proof is complete 0
The following shows that the converse of Lemma 41 also holds
Lemma 42 H the group G has property (52) then the trace tr is faitllful on M~(G)
Proof Suppose G has property (52)
Assume that the trace tr is not faithful on M~(G) Then there exists ltp E Mg(G) O
such that tr(cpcp) = O By the assumption of property (52) u(ltp) =f 0 Let Xo E u(cp)
Let cpn = EZEG chz Espan EA(G) and cpn -+ cpo By Lemma 31 limn _ oo cO =f O Thus
= lim Ic~ 12 n oo~ zEG
~ lim Ic~OI2 gt 0n-+oo
contradicting that tr(cpltp) = O Therefore tr is faithful on M~(G) 0
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
23 SPECTRUM AND AMENABILITY
Corollary 43 Let G be locally compact group Then G has property (82 ) if and only
if the trace tr is faithful on M~(G)
To give another characterization of property (S2) we need the following preparation
Recall that Gd donotes the algebraic group G endowed with the discrete topology
The M~(~) is the reduced C-algebra of Gd Dunk and Ramirez in [9 Theorem
21] showed that IIJlIIM~(Gd) ~ IIJlIIM~(G) for all Jl E Md(G) Thus the map Jl 1-+ Jl
Jl E Md(G) = Md(Gd) extends a C-homomorphism r of M~(G) onto M~(~) A
natural question is when r is a C-isomorphism (or M~(G) M~(~raquo Applying
Dunk and Ramirez [9 Theorem 23] we answer this question in the following
Lemma 44 Let G be a locally comact group Then M~(G) M~(~) if and only if
the trace tr is faithful on M~(G)
Proof Dunkl and Ramirez in [9] used Tr to denote the finite trace on M~(~) defined
by Tr(Jl) = Jl(e) Jl E M(Gd) = Md(Gd) = Md(G) Then Tr is continuous because it
is also the restriction of a topologically invariant mean Dunkl and Ramirez proved that
Tr is always faithful on M~(~) [9 Theorem 23])
Now suppose M~(G) M~(~) Then tr = Tr on M~(G) and hence tr is faithful
Conversely suppose tr is faithful on M~(G)
First we observe that Tr(rcp) tr(cp) for all cp E M~(G) In fact if Jl E Md(G) then
r(Jl) = Jl and hence Tr(rJl) = Jl( e) = tr(Jl) The assertion follows from the continuity
of Tr and tr
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
ZHIGUO HU24
Let tp E M~(G) a Then
Tr( (ftp)(ftpraquo = Tr(f( tpraquo = tr( tp tp) gt o
Thus ftp =1= o It follows that r M~(G) - M~(~) is injective and hence is a
C-isomorphism 0
By Corollary 43 and Lemma 44 we are ready to present one of the main results of
this paper
Theorem 45 Let G be a locally compact group Then the following assertions are
equivalent
(1) G has property (S2)
(2) The trace tr is faithful on M~(G)
(3) M~(G) ~ M~(~)
In particular G has property (S2) if either G is discrete or Gd is amenable
Next we consider property (Sp) for general 1 lt p lt 00 Let G be amenable and
1 lt P lt 00 Herz showed that the identification of functions gives a contraction A(G) shy
Ap(G) dually there is a contraction Ap(G) - A(G) (see Herz [16]) In this case
Mp(G) ~ M 2(G) M~(G) ~ M~(G) etc Hwe use Op(tp) to denote the norm spectrum
of tp in Ap(G) then Op(tp) ~ 02(tp) for all tp E M(G) O ~ M~(G) O since
A(G)nCoo(G) is IImiddot II A (G)-dense in Ap(G) We are unable to conclude Op(tp) =1= 0
directly from 02 ( tp) =1= 0 However using the above two constractions and an argument
similar to that for Lemma 41 we can also prove the following
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
25 SPECTRUM AND AMENABILITY
Lemma 46 Let G be an amenable locally compact group and 1 lt p lt 00 If G has
property (S2) then G has property (Sp)
Proof Suppose G has property (S2) Then by Lemma 42 the trace tr is faithful on
M~(G)
Let ip E M(G) O and ipn E span EA(G) such that ipn -+ ip in the 1 IIA(G)middot-norm
For the same reason we may assume that there exists a compact subset K of G such
that supp ip s K and supp ipn ~ K for all n We may also assume that 1Iip1l Ap(Gt = 1
d ~
and lIipnIlA(G)middot = 1 for all n Then ip E M2(G) O cpn E spanEA(G) ipn -+ ip in the
lIIIA(Gt-norm IIcpIlA(G)middot s 1 and lIipnIlA(Gt s 1 for all n
Let U a and gn be the same functions in A(G) as in the proof of Lemma 41 Then
lIualAG) = ua(e) = 1 and IIgnIlA(G) s IIgnIlA(G) s 1 Since ip E APp(G) and
ipnmiddot gn - ip gn -+ 0 in the IImiddotIIA(G)middot-norm we may assume that ipnmiddot gn -+ T E M(G)
in the II middotlAp(G)middot-norm Thus Pn gn -+ T E M~(G) in the IIA(G)middot-norm According
to the proof of Lemma 41 T E M(G) nM~(G) O and T has a nonzero discrete part
So by Lemma 313(a) up(T) 0 But ip gn -+ T in the IImiddotIIA(G)middot-norm By definition
up(T) s up(ip) It follows that up(ip) 0 Since ip E M(G) OJ is arbitray the group
G has property (S) 0
Finally we would like to discuss the relation between the property (Sp) and the
amenability of G As mentioned in the introduction Bedos showed that Gd is amenable
iff G is amenable and M~(G) ~ M~(G) (see [1 Theorem 3]) Combining this result
with Theorem 45 Lemma 46 and the paragragh before Lemma 46 we can conclude
the following
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
26 ZHIGUO HU
Theorem 47 Let G be locally compact group Then the following assertions are
equivalent
(1) Gd is amenable
(2) G is amenable and G has property (S2)
(3) G is amenable and G has property (Sp) for some 1 lt p lt 00
(4) G is amenable and G has property (Sp) for all 1 lt p lt 00
Remark 48 (i) From Theorem 45 we see that the Fourier algebra A(G) has property
(S) if G is either discrete or amenable as a discrete group We do not know whether the
converse is also true In other words we do not know whether there is no non-amenable
nondiscrete group G with property (S2) (or equivalently M~(G) ~ M~(~)) If this is
the case we would have the following nice result
For any nondiscrete locally compact group G A(G) has property (S) if and only if
Gd is amenable if and only if the trace tr is faithful on M~(G) if and only if M~(C1)
M 2d-(Gd)
(ii) Recall the property (A) mentioned in Remark 38(ii) Chou Lau and Rosenblatt
[5J proved among other characterizations that an infinite compact gruop G has property
(A) iff M~(G) nPF2 (G) = OJ For any nondiscrete locally compact group G the group
G has porperty (S2) implies M~(G)nPF2(G) = OJ (by Corollary 314(f)) Meanwhile
it is possible that G is compact M~(G)npF2(G) = OJ and G fails to have property
(S2) (hence Gd is not amenable) See [5 Remark I4J for such groups G Therefore the
converse of Corollary 314( f) is not true
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
27 SPECTRUM AND AMENABILITY
(iii) There was a gap in the proof of [24 Proposition 54] The scalar there is equal
to (ltp m) However we were unable to draw that = 1 or f O It is seen now that
Proposition 54 of [24] may not hold if Gd is not amenable
Since any abelian group is amenable as a simple application of Theorem 46 we have
the following
Corollary 49 Let G be a locally compact abelian group Tben G bas property (Sp)
for all lt p lt 00
Proposition 410 Let G be locally compact group witb property (S2) Tben each
Proof Suppose G has property (S2) By Theorem 45 M~(G) ~ M~(Ga) Therefore
the series ltp = EZEU(IP) (ltp mz )6z is convergent in M~(G) Let
p = ltp - L (ltp mz)6z zEu(IP)
Thenp E M~(G) and by Lemma 31 (p m z ) =0 for all x E G According to Lemma
24 u(p) =0
Since G has property (S2) it follows that p = 0 that is rp =EZEu(IP)(ltP m z )6z bull The
proof is complete 0
Corollary 411 Let G be locally compact group witb property (S2) Tben for any
ltp E M~(G) tbere exists a sequence (Un)n~l in A(G) such tbat ltp bull Un -+ ltp in tbe
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
28 ZHIGUO HU
Proof Suppose G has property (S2) Let cp E Mg(G) Then u(cp) is a countable subset
of G say u(cp) = (X n )nl Let Un E A(G) such that u(Xt) = 1 for all k $ n and
U(Xk) = 0 for all k gt n Then
cP bullUn = L (cp mxSxbull k~n
Therefore according to Proposition 410 cp Un -+ cp in the IImiddotIIM~(G)-norm The proof
is complete 0
We conclude this paper with the following immediate consequence of Corollary 314(e)
and Theorem 45
Corollary 412 Let G be a second countable locally compact group and 1 lt p lt 00
H G is either discrete or amenable as a discrete group then a proper closed ideal [ of
Ap(G) is synthesizable if and only if I = lip for some cp E M~(G) OJ
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
[7] J Dixmier C-algebra3 Amsterdam North-Holland 1977
[8] C De Vito Characterization3 of those ideal3 in Ll (JR) which can be 3ynthe3ized
Math Ann 203 (1973) 171-173
[9] C F Dunkl and D E Ramirez CmiddotalgebTa3 generated by Fourier-Stieltje3 tran3shy
formationJ Trans Amer Math Soc 164 (1972) 435-441
[10] C F Dunk and D E Ramirez Weakly almost periodic functionals on the Fourier
algebra Trans Amer Math Soc 185 (1973) 501-514
[Il] P Eymard Lalgebra de Fourier dun groupe localement compact Bull Soc Math
France 92 (1964) 181-236
[12] E E Granirer On 30me space3 of linear functional3 on the algebra3 Ap(G) for
locally compact groups Colloq Math 52 (1987) 119-132
[13] E E Granirer On convolution operators which are far from being convolution by
a bounded measure Exp03itory memoir C R Math Rep Acad Sci Canada 13
(1991) 187-204
[14] E E Granirer On convolution operator3 with small support which are far from
being convolution by a bounded mea3ure Colloq Math 67 (1994) 33-60
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
SPECTRUM AND AMENABILITY 29
REFERENCES
[1] E Bedos On the C-algebra generated by the left tran3lation of a locally compact
group Proc Amer Math Soc 120 (1994) 603-608
[2] M Bekka A T Lau and G Schlichting On invariant 3ubalgebra3 of the Fouriershy
Stielje3 algebra of a locally compact group Math Ann 294 (1992) 513-522
[3] M Bekka and A Valette On dual3 of Lie groUp3 made discrete J Reine Angew
Math 439 (1993) 1-10
[4] C Chou Alm03t periodic operator3 in V N(G) Trans Amer Math Soc 317
(1990) 229-253
[5] C Chou A T Lau and J Rosenblatt Approximation of compact operator3 by
3UmJ of tranJlation3 lllinois J Math 29 (1985) 340-350
[6] M G Cowling and J J F Fournier Inclusions and noninclusion3 of 3pace3 of
convolution operators Trans Amer Math Soc 221 (1976) 59-95
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30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
30 ZHIGUO HU
[15] F Greenleaf Invariant Means of Topological Groups and Their Applications Van
Nostrand Math Studies no 16 Van Nostrand New York 1969
[16] C Herz The theory of p-spaces with an application to convolution operators Trans
Amer Math Soc 154 (1971) 69-82
[17] C Herz Harmonic synthesiJ for subgroups Ann Inst Fourier 23 (1973) 91-123
[18] E Hewitt and K A Ross Abstract Harmonic Analysis Vols I II Springer Verlag
New York 1970
[19] Y Katznelson An Introduction to Harmonic Analys Dover Publications Inc
New York 1976
[20] A T Lau Uniformly continuous functionals on the Fourier algebra of any locally
compact group Trans Amer Math Soc 251 (1979) 39-59
[21] A L T Paterson Amenability Amer Math Soc Providence Rhode Island
1988
[22] J P Pier Amenable Locally Compact Groups John Wiley and Sons New York
1984
[23] P F Renaud Invariant means on a class of von Neumann algebras Trans Amer
Math Soc 170 (1972) 285-291
[24] A Ulger Some results about the spectrum of commutative Banach algebras under
the weak topology and applications Mh Math 121 (1996) 353-379
[25] G Zeller-Meier Representations fideles des produits croises C R Acad Sci Pairs
Ser A 264 (1967) 679-682
