Math. Proc. Camb. Phil. Soc. (1994), 115, 39 3 9

Printed in Great Britain

Minimal primal ideals in Banach algebras

BY DOUGLAS W. B. SOMERSETThackit Eaves, Highclere, Newbury, Berkshire, RG15 9QU

(Received 19 February 1993; revised 5 May 1993)

Abstract

An ideal / in a ring R is primal if whenever Jx,.,. ,Jn is a finite set of ideals of Rwith J1...Jn = {0} then Jt £ / for at least one i e {1, . . . , n}. If the ring is commutativethen is is easily shown that each primal ideal contains a prime ideal. In this paper itis shown that in a separable semi-simple Banach algebra each primal ideal containsa prime ideal, and that the space of minimal prime ideals is compact and extremallydisconnected in the hull-kernel topology. An example is given of an inseparable C*-algebra with a primal ideal which does not contain a prime ideal.

1. Introduction

We start with some definitions. An ideal / in a commutative ring it! is(i) prime if whenever a,beR with abel then either ael or bel,

(ii) pseudoprime if whenever a,beR with ab = 0 then either ael or bel.(iii) universally pseudoprime if whenever ax,... ,aneR(l < n < oo) with

« j . . . an = 0 then atel for at least one ie{l,..., n}.It is trivial that each prime ideal is universally pseudoprime, and that each

universally pseudoprime ideal is pseudoprime. It is also easy to see that each idealwhich contains a prime ideal must be universally pseudoprime. Conversely, a well-known argument of Krull's shows that each proper universally pseudoprime idealcontains a prime ideal [11], [15; 2-5]. There are simple examples of pseudoprimeideals which are not universally pseudoprime [11]. Example 21 of [18] is a semi-prime, Noetherian ring in which the Jacobson radical is pseudoprime but notuniversally pseudoprime. On the other hand, it is known that when X is a completelyregular topological space each pseudoprime ideal in CR(X) (the ring of continuousreal-valued functions on X) is universally pseudoprime [13], [11].

The purpose of this paper is to explore the analogous situation for non-commutative rings, and for Banach algebras in particular. First we give theappropriate definitions. An ideal / ('ideal' in this paper means 'two-sided ideal') inan (arbitrary) ring R is:

(i) prime if whenever J and K are ideals of R with JK £ I then either J £ / orK^I,

(ii) 2-primal if whenever J and K are ideals of R with JK = {0} then either J £ /or K £ /,

(iii) primal if whenever Jlt..., Jn (1 < n < oo) are ideals of R with Jl...Jn = {0}then Jt £ I for at least one ie{l,..., n).

40 DOUGLAS W. B. SOMERSET

If the ring R is commutative it is well known that our two definitions of ' prime'coincide [15; 2*1]. Similarly it is easily checked that, in this case,'pseudoprime' is thesame thing as '2-primal' and "universally pseudoprime' is the same as 'primal'.

As before, it is trivial that prime ideals are primal and that primal ideals are 2-primal. Any ideal which contains a primal ideal is primal. A simple application ofZorn's Lemma shows that each primal ideal contains a minimal primal ideal, so eachproper primal ideal contains a prime ideal if and only if each proper minimal primalideal is prime.

The motivation comes partly from the theory of closed primal ideals in C*-algebras. Initially these were studied because of their connexion with limits offactorial states [2] but they are now objects of interest in their own right [1], [3], [4].Particular attention has been paid to MinPrimal (A), the space of minimal closedprimal ideals, with the lower topology (see Section 2). If A is a separable C*-algebrathen Min Primal (A) has a dense subset consisting of closed prime ideals [1; 4-6], butit is easy to give examples showing that not every minimal closed primal ideal isprime. It might be, however, that each minimal closed primal ideal contains a non-closed prime ideal. It turns out that they do when the C*-algebra is separable (Section4), but not necessarily when the algebra is inseparable (Example 3-7). In fact, it isshown in Section 4 that if A is a separable, semi-simple Banach algebra, then eachminimal primal ideal of A is prime. Furthermore the space of minimal primal idealsof A, with the hull-kernel topology, is compact and extremally disconnected (thatis, the closure of every open set is open). In Section 3 the same is shown for vonNeumann algebras.

The commutative examples already mentioned show that 2-primal ideals are notnecessarily primal. It is natural to wonder what happens in non-commutative C*-algebras (being the non-commutative analogues of C(X), at least for X compact) butthe example in [2] page 59 shows that even here 2-primal (indeed 'n-primal') idealsneed not be primal. Nevertheless we show in Section 3 that for a large class of C*-algebras, called 'quasi-standard C*-algebras', each 2-primal ideal is primal. Thisclass includes von Neumann algebras and C*-algebras with Hausdorff primitive idealspace.

2. Preliminaries

In this section we assemble some useful facts about primal ideals in rings andlattices. We begin with some definitions and results from lattice theory.

Let L be a lattice with minimal element 0. Recall that a subset / of L is an idealif it satisfies

(i) xel, yeL, y ^ x=>yel, and(ii) x,yeL=>xW yeL.

Let Id(L) denote the complete lattice of ideals of L. An element p of a lattice L is saidto be prime if whenever a,beL with aAb^p then either a ^ p or b ^ p, and to beprimal if whenever xx,..., xn eL with xx A ... A xn = 0 then xi ^ p for at least onei e {1,..., n}. An ideal / in L is prime (respectively primal) if it is a prime (respectivelyprimal) element of the lattice Id(L). It is easily checked that / is prime if and onlyif whenever x,yeL with x Ay el then either xel or yel. Similarly/ is primal if andonly if whenever xlt ...,xneL with x1A. . .Axn = 0 then xtel for at least oneie{l, ...,n}. Let Spec(i/) denote the set of prime ideals of L and let MinSpec(L)

Minimal primal ideals in Banach algebras 41denote the set of minimal primal ideals oiL, both equipped with the restriction of thelower topology of L (defined below). A simple application of Zorn's Lemma showsthat each primal ideal contains a minimal primal ideal, so Min Spec(£) is non-empty.

The first proposition is well-known.

PROPOSITION 21 . Let L be a distributive lattice and let P be a primal ideal ofL. IfPis a minimal primal ideal then P is prime. Furthermore the following are equivalent:

(i) P is a proper minimal primal ideal,(ii) if aeL then aePothere exists b$P such that a A b = 0,

(iii) if aeP there exist elements blt... ,bneL such that bt$P(l ^ i < n) andaA&! A ... t\bn = 0.

Proof. The fact that minimal primal ideals are prime, and the implication (i) => (ii),are well-known. They can easily be deduced from [22] 1-2-4 for example.

(ii) => (iii). This is trivial.(iii) => (i). Suppose that J is an ideal properly contained in P and let aeP\J. By

assumption there are elements b1,...,bn of I such that 61 ; . . . , fen$P anda Ab1 A ... Abn = 0. Then a$ J and bt$J(1 ^ i ^ n), so J is not primal. I

We have stated the weak form (iii) above because this is useful later on when wewant to check that certain ideals are minimal primal.

Definition. If L is a lattice the lower topology on L is the topology which has as asub-base the collection of sets

{{leL:l%x}:x€L}.

The restriction of the lower topology of L to the set of prime elements is the well-known hull-kernel topology.

The next proposition is taken from [21]. Part (ii) extends [1], proposition 3-1 fromC*-algebras to the general case. [1] proposition 3-1 was one of the foundationalobservations for the whole subject.

PROPOSITION 2-2. Let L be a complete lattice.(i) The set of primal elements of L is closed in the lower topology.

(ii) IfS is a set of primal elements ofL with iniS = 0 then the closure ofS in the lowertopology is equal to the set of all primal elements of L.

(iii) If S and T are sets of prime elements of L with infS = inf T then S and T havethe same closure in the lower topology.

Definition. If R is a ring let Id(R) denote the lattice of all ideals of-ft, equipped withthe lower topology. (Recall that the meet of two ideals in this lattice is theirintersection and not their product.) Let Spec (.ft) denote the set of prime ideals of Rwith the hull-kernel topology.

(It would be natural to define Primal (R) to be the set of all primal ideals of R butthe name is already in use for the set of closed primal ideals of a C*-algebra.)

The product of two ideals is contained in their intersection, so it follows that eachprime ideal of R is a prime element of the lattice Id(R) and that each primal ideal isa primal element. The converses are not true: there are elementary examples of idealswhich are not primal but which are prime elements in the lattice. However, if R is

42 DOUGLAS W. B. SOMERSET

a semi-prime ring (that is, a ring with no non-zero nilpotent ideals) then if/ 1 ; . . .,IneId(R) with I1...In = {0} then (/x n ... fi In)

n = {0}, so Ix n ... n /„ = {0}. Itfollows that an ideal of R is primal if and only if it is a primal element of ld(R).

Proposition 2-2 (ii), and the remarks above, imply that if R is a semi-prime ring thenthe set of primal ideals of R is precisely the closure of Spec (ii!) in the lower topology.(Recall that an alternative definition of semi-primeness is that the intersection of theprime ideals is zero.) It would be interesting to know if the same is true for all rings.If it were then it would follow that every primal ideal of a ring contains the primeradical of the ring (that is the intersection of all the prime ideals). This is true forBanach algebras because the prime radical of a Banach algebra is generated by thenilpotent ideals [7] and a primal ideal must contain every nilpotent ideal.

Definition. If A is a ring let Min Spec (A) denote the set of minimal primal ideals ofA equipped with the lower topology from Id(A). If A is a Banach algebra letMin Primal (A) denote the set of ideals which are minimal in the class of closed,primal ideals, also equipped with the lower topology from Id(A).

(Min Primal (A) could equally well be equipped with the lower topology from thelattice of closed ideals of A, which would be the same topology.)

When A is a C*-algebra the space Min Primal (̂ 4) has been an object of study [1],[3], [4] but the author is not aware of any work on the space for Banach algebras.If A is a C*-algebra with Hausdorff primitive ideal space then every closed primalideal of A is maximal; see [2] page 60. The same is true if A is a strongly regularcommutative Banach algebra [10], but it is not true for all regular commutativeBanach algebras. There is an example in [9] of a regular, uniform, commutativeBanach algebra with a minimal closed primal ideal which is not prime.

If A is a commutative ring then Min Spec (̂ 4) is simply the space of minimal primeideals, studied by many authors. Perhaps some justification is needed for using thename Min Spec (̂ 4) in the non-commutative case, when minimal primal ideals neednot be prime (see Example 3-7). There are two reasons for doing this. The first istheoretical, namely that what seems to be important about Min Spec (̂ 4), even forcommutative rings, is that it is the set of minimal elements of the closure of the setof prime ideals in the hull-kernel topology (at least, this is true for semi-prime rings,see above). This is why the hull-kernel and patch topologies coincide on Min Spec (̂ 4)([19], [20], [21]) and this is why the minimal prime ideals of a semi-primecommutative ring have the useful characterization in terms of annihilators, [24], see[23], [21] and Theorem 2-3. The second reason is pragmatic: it doesn't seem thatanybody will be inconvenienced by our definition because in every case known to theauthor when minimal prime ideals have been used in non-commutative rings, theyhave turned out to be the minimal primal ideals. This is true in Noetherian rings [15],[25], in spectral rings [5] (of which Noetherian rings are a special case), in reducedrings [26], [31], and in Rickart *-rings [32].

Let R be a ring. For aeR let /„ denote the smallest ideal of R containing a. HI isan ideal of R let I1 denote the largest ideal of R satisfying IIL = ILI = {0}.

The next theorem is basic to the whole subject. The proof (of a more general resultfor lattices) is to be found in [23].

THEOREM 2-3. Let R be a semi-prime ring and let P be a primal ideal of R. Thefollowing are equivalent:

Minimal primal ideals in Banach algebras 43

(i) P is a proper minimal primal ideal,(ii) if I is a finitely generated ideal of R then I £ P if and only if there exist a finite

number of ideals I1,...,In of R such that Ix, . . . , / „ $ P and II1...In = {0},(iii) if aeR then aeP if and only if there exist a finite number of ideals I1,.. . , / n of

R such that Il7 . . . , / „ $ P and IaIx . . . /„ = {0}.

Proof, (i) => (ii). This is a special case of [23] theorem B and lemma 3-6.(ii) => (iii). This is trivial.(iii) => (i). Suppose that J is an ideal of R properly contained in P and let aeP\J.

By assumption there are ideals I1,...,In of R such that / 1 , . . . , / _ $ P andIaI1 ...In = {0}. Then / a $ J and lt $ J ( l ^ i ^ n), so J is not primal. I

If R is a commutative ring and P is minimal primal then P is prime, so (iii) ofTheorem 23 can be strengthened as follows [24]:

(iii)' if aeR then aeP if and only if 1^ $ P.One useful consequence of Theorem 2-3 is that if P is a minimal primal ideal in a

semi-prime ring and if a eP then I^1 ^ P. When J? is a Banach algebra, I^1 is a closedideal. This means that we shall be able to use analytic techniques to investigateminimal primal ideals in semi-prime Banach algebras.

3. Quasi-standard C*-algebras

In this section we identify the minimal primal ideals in a special class of C*-algebras called quasi-standard C*-algebras. These resemble commutative C*-algebrasin many ways and we show that the study of their minimal primal ideals can bereduced, for many purposes, to the study of minimal prime ideals in commutativeC*-algebras. In particular we use the commutative theory to show that each 2-primalideal is primal in a quasi-standard C*-algebra with identity. AW*-algebras are quasi-standard and we show that each minimal primal ideal is prime in an AW*-algebraand that the space of minimal prime ideals is compact and extremally disconnected.Finally, we give an example of a von Neumann algebra with a primal ideal which isnot prime and of a quasi-standard C*-algebra with a minimal primal ideal whichis not prime.

The basic technique is to map the ring onto a distributive lattice in such a way thatthe minimal primal ideals are mapped onto the minimal prime ideals of the lattice.This idea of using distributive lattices to study ideals of rings has been exploited bymany authors [24], [12], [29] and [5]. It goes back, ultimately, to Stone's work onBoolean algebras.

If X is a topological space let C(X) and Gb(X) denote the algebras of continuouscomplex-valued functions and bounded, continuous complex-valued functions on X,respectively.

Definition. Let A be a C*-algebra. Then A is said to be quasi-standard if thefollowing are true:

(i) Min Primal (̂ 4) is a locally compact, Hausdorff space,(ii) for all as A the function P-* ||a + P|| (PeMin Primal (A)) is continuous and

vanishes at infinity,(iii) if/eC'6(MinPrimal(.4)) and aeA there exists beA such that f(P)(a+P) =

b + P (PeMin Primal {A)).

44 DOUGLAS W. B. SOMERSET

Remarks. The continuity requirement of (ii) is satisfied by every C*-algebra [1; 4-3].I t can be shown that (iii) implies (i) and (ii). There are a number of equivalentdefinitions of quasi-standardness, see [3; 3-3, 3-4].

The class of quasi-standard C*-algebras is reasonably large. It includes allcommutative C*-algebras, all C*-algebras with Hausdorff primitive ideal space, allvon Neumann and AW*-algebras, and a number of group C*-algebras.

Let A be a Banach algebra. For a eA let coz (a) = {P e Min Primal (A): a 4 P} and letsupp (a) be the closure of coz (a) in Min Primal (A). Note that

/£ = {b eA: coz (a) (] coz (6) = 0 } .

LEMMA 3*1. Let A be a quasi-standard C*-algebra and let L(A) denote the collection{coz (a): aeA}. Then L(A) is a distributive lattice.

Proof. Let a,beA. We wish to show that coz (a) U coz (b) and coz (a) D coz (6)belong to L(A). Since coz (a) U coz (b) = coz (a*a + 6*6), coz(a) U coz(6)eL(^4). Let/eC6(Min Primal {A)) such that/(P) = | |6+P|| (Pe Min Primal (A)) and let ceA suchthat c + P =/(P) (a+P) (Pe Min Primal^!)). Then coz(c) = coz(a) fl coz(6). Hencecoz (a) fl COT. (b) eL(A). I

THEOREM 3-2. Let A be a quasi-standard C*-algebra. Let @:A-+L(A) be definedby Q(a) = coz (a) (aeA). Then 0 induces a homeomorphism, also called 0, fromMin Spec (A) onto Min Spec (L(A)).

Proof. Let P6Min Spec (A). If a, beP then coz (a) U coz (6) = coz(a*a + 6*6)e0(P).If aeP and beA with coz (6) £ coz(a) then bel^1 £ P, by Theorem 2-3. Hencecoz(6)6 0(P). This shows first that 0(P) is an ideal of L(A) and secondly thatP = 0-!(0(P)).

Next we show that 0(P) is a primal ideal inL(A). Suppose that ateA (1 < i < n)andcoz (a,) ̂ 0(P). T h e n a ^ P (1 ^ i ^ n), so / a i / a s . . ./On # {0}. Le t6e / a i / a 2 . . ./On\{0}.Then 0 #= coz (6) £ D™-1cos(a<). Hence 0(P) is primal.

Next we show that 0(P) is a minimal primal ideal inL(^4). Suppose that aeA withcoz(a)e©(P). Then aeP so by Theorem 2-3(iii) there exist bv ...,bneA such that7 6 j $ P ( l ^ t ^ n ) and IaIbi. ..Ibn = {0}. But this implies that coz (64) <£ 0(P)(1 ^ i < n) and that coz (a) n nf_x coz (bt) = 0 . Hence 0(P) is minimal primal byProposition 2*1 (iii).

I t is clear from the fact that P = 0-1(0(P)) that 0 is an injective map fromMin Spec (A) to Min Spec (L(A)). We now prove that 0 is a surjective map. Let#eMinSpec(L(.4)) and define P = Q^iQ) = {aeA : coz(a)eQ}. Then P is an ideal ofA and 0(P) = Q. Suppose that Ix,... ,In are ideals of A with I1,...,In $ P. For eachie{l,...,n} let ateIt\P. Then coz(«;)£# s° ri"_1coz(ai) 4= 0 , since Q is primal. Itfollows that {0} + Ia .. .Ia £ 7X.. ./„. Hence P is primal. If P' is a minimal primalideal contained in P then 0(P') £ 0(P) = Q. But 0(P') is minimal primal, by theprevious paragraph, so 0(P') = 0(P) and hence P is minimal primal. This shows that0 is a bijection from Min Spec (̂ 4) onto Min Spec (L(A)).

Finally we show that 0 is a homeomorphism. Let K be an ideal in L(A) and let Jbe the ideal of A defined by J = Q~l(K). Then, for P e Min Spec (A),P£J if and onlyif 0(P) $ Q(J) = K. Hence

{Pe Min Spec (.4): P $ J] = Q-X({Q e Min Spec (L(A)): Q $ K})

Minimal primal ideals in Banach algebras 45and both sets are open in the respective lower topologies. Since the collection of sets

{{Q e Min Spec (L(.4)): Q $ K}: K e Id(L(A))}

forms a base for Min Spec (L(A)) (because all the elements of Min Spec (L(A)) areprime, Proposition 2-1) it follows that 0 is continuous.

For each a eA set Ja = f){P eMin Spec (A): a eP}. Then it is clear that thecollection of sets

{{Pe Min Spec (.4): P $ Ja}: aeA}

is a sub-base for the topology of Min Spec (.4). For each aeA setKa = {coz (b): beJa}.It is easily checked that Ka = D{0(P): P 2 Ja}, so Ka is an ideal of L(A). Since

0({PeMin Spec (^) :P $ JJ) = {QeMmSpec (L(A)): Q 3>Ka},

an open set, and since 0 is a bijection, it follows that 0 is open. Hence 0 is ahomeomorphism. I

When A is a quasi-standard C*-algebra with an identity there is a close connectionbetween the minimal primal ideals of A and those of the centre Z(A) of A. In this caseMin Primal (A) is compact and the map z^-z + P(zeZ(A),Pe Min Primal (A)) definesan isomorphism between Z(A) and C(MinPrimal (A)), [3] page 351 and theorem 3-3).Thus if A is a quasi-standard C*-algebra with an identity we can identify Z(A) withC(MinPrimal (.4)), and hence identify L(A) with L{Z(A)).

THEOREM 3-3. Let A be a quasi-standard C*-algebra with an identity and let Q.be the map P-+P n Z(A) (Pe Min Spec (̂ 4)). Then Q defines a homeomorphism fromMinSpec(A) ontoMinSpec(Z(A)) and Q.'1 is given by Q.-1(Q)=AQ(QeM.inSY>ec(Z(A))).

Proof. For this proof, let 0^ : MinSpec(vl)->-MinSpec(Zv^4)) be the map obtainedwhen Theorem 3-2 is applied to A and let QZ(A): Min Spec (Z(A))-> Min SpecL(A))be the map obtained when Theorem 3-2 is applied to Z(A). (Recall that weidentify L(A) and L(Z(A)).) Set Q = ®ZIA)°®A- Then Q is a homeomorphism fromMinSpec(^) onto Min Spec (Z(A)), and it is easy to check that Q(P)=P()Z{A),for P e Min Spec (.4). It is also clear that if Q e Min Spec {Z(A)) and P = Q'^Q) thenQ c p . Hence AQ c p.Let aeP and let zeZ(A)+ such that z+R = \\a+R\\ foreachi?eMinPrimal (A). Then coz (z) = coz (a)e0(P) so zeP. Hencez'eQ = P f! Z(A).But the element

- ( 1 .can be denned by functional calculus, and a — bz*, so aeAQ. Thus P £ AQ, soP = AQ. I

COROLLARY 3-4. Let A be a quasi-standard C*-algebra with an identity. If I is a 2-primal ideal of A then I is primal.

Proof. Let Z denote the centre of A. First note that J = 10 Zsa is a pseudoprimeideal in the real algebra Zsa (the set of self-adjoint elements of Z). For if/, g e Zsa withfg = 0 then Af and Ag are ideals of A with (Af) (Ag) = {0}. At least one of Af and Agis contained in / , because / is 2-primal, so at least one of/ and g is in J. This showsthat J is pseudoprime. As mentioned in the introduction, it follows from [13] that«/is universally pseudoprime in Zsa.

46 DOUGLAS W. B. SOMERSET

N o w l e t y1,...,yneZ w i t h y l ...yn = 0. F o r e a c h ie{l,...,n} l e t zt = \yt\. T h e nz1 z2 ... zn = 0, so z| z | . . . z?n = 0. Since J is universally pseudoprime z'e J for at leastone ie{l ...n). Set

?

Then 2/4 = z\wt, so t/4e/ n Z. Hence / 0 Z is universally pseudoprime in Z. Let Q bea minimal primal ideal of Z contained in / D Z. Then AQ is contained in / and AQ isprimal, by Theorem 3-3. Thus / contains a primal ideal, so I is primal. I

As we mentioned before, there is an example of a C*-algebra with a 2-primal idealwhich is not primal in [2] page 59. The ideal Q1 f) Q2 (1 Q3 in [1] 4-12 is also 2-primalbut not primal. The assumption that A has an identity in Corollary 3"4 is notnecessary. The proof, however, in the case when A does not have an identity involvesembedding A in a quasi-standard C*-algebra with an identity, and the details are toolengthy to include.

The next theorem sounds like a theorem about closed ideals, so we remind the readerthat the ideals under discussion are not necessarily closed.

THEOREM 3'5. If A is an AW*-algebra then Min Spec (yl) is compact and extremallydisconnected and each minimal primal ideal of A is prime.

Proof. Since A is quasi-standard Z(A) is isomorphic to C(Min Primal (A)) and Z(A)is an AW*-algebra, so MinPrimal (A) is extremally disconnected. It follows fromTheorem 3-3 and [19] Theorem 5-3 that Min Spec (A) is compact and extremallydisconnected.

Extremally disconnected spaces are F-spaces, so it follows from Theorem 3-3 and[12; 14-25] that if P is a minimal primal ideal of A there exists a point Q inMin Primal (A) such that P = {a eA : Q $ supp (a)}. (Recall that supp (a) is the closureof coz (a).) Let a, beA with a,b$P. In order to show that P is prime it is sufficientto find ceA such that acb$P [15; 2-1]. Since Q(P) is a prime ideal in L(A),COT. (a) fl coz (b) £ Q(P), that is, Q is in the closure of coz (a) f] coz (b). It is thereforesufficient to find ceA such that coz (acb) is dense in coz (a) fl coz (6), for thenQesupp(acfe), so acb$P.

Let R be any point in coz (a) 0 coz (6). Then .ft is prime, by the proof of [14] Lemma11, so there exists a projection eeAsa such that aeb$R. Let N be a clopenneighbourhood of R contained in coz (aeb), and let 2 be a central projection withcoz (2) = N. Replacing e by ze, we may assume that coz (e) = coz (aeb). Now let ea bea maximal family of projections in Asa with the following properties:

(i) the eas have pairwise orthogonal central covers [6] §6-1,(ii) for each a, coz (ej = coz (aeab).

Set c = 2 a e a . Condition (i) ensures that if i?ecoz(ej for some a then Re coz (aeb).Condition (ii), together with the maximality of the family ea, ensures that coz (acb)is dense in coz (a) D coz (b). Hence acb$P. I

As a matter of fact it follows from Theorem 3-3 and [19], theorem 5-3 that whenA is an AW*-algebra Min Spec (̂ 4) is homeomorphic to Min Primal (A). If yl is aseparable quasi-standard C*-algebra with an identity then Theorem 3-3 and [19],theorem 5-6 imply that Min Spec (yl) is again compact and extremally disconnected,

Minimal primal ideals in Banach algebras 47but we prove a more general result in Theorem 4-3. There are examples in [19]showing that even when A is a commutative C*-algebra MinSpec(^4) need not beeither compact or extremally disconnected. It need not even be basicallydisconnected [8]. An alternative proof of the compactness and extremal dis-connectedness of MinSpec(^4), when A is an AW*-algebra, is given in [21].

The proof of Theorem 3-5 could be modified to show that if A is a separable quasi-standard C*-algebra with an identity then each minimal primal ideal of A is prime,but we prove a more general result in Theorem 4-3.

In a commutative AW*-algebra every primal ideal is prime [12; 14-24—26], and ina non-commutative AW*-algebra every closed primal ideal is prime, by the proof of[14], lemma 11. We shall therefore give an example of a primal ideal in an AW*-algebra which is not prime.

Example 3-6. A von Neumann algebra with a primal ideal which is not prime.Let A be a von Neumann algebra and let J be the strong radical of A, that is, the

intersection of the maximal ideals of A. Assume that A is of Type IOT, so that J is non-zero [17], proposition 2-3. Let n be the canonical map n:A-^A/J. The primitiveideal space of A/J is Hausdorff and extremally disconnected, [17], proposition 2-3,and [16], page 911. Assume thati? is a non-isolated point of this space. If we assumethat A is countably decomposable then R is not minimal prime [12; 7-15, 12H], so Rcontains a (unique) non-closed, minimal prime ideal P. Set Q = n~1(P) and S = v~1(R).Then Q is a non-closed ideal in A containing J, and S is a maximal ideal in A. By [17],proposition 2-3 there exists a minimal closed primal ideal G in A such that S = G + J.Note that Q $ G, because Q $ J but Q =t= S, and G $ Q, because Q is dense in S andG 4= S [16], theorem 3. Hence the ideal G 0 Q is not prime. But

Let H be a minimal prime ideal contained in G. The description of H given in theproof of Theorem 3-5 shows that H is the smallest ideal which is dense in G. HenceG fl Q 2 H, so G n Q is primal.

Example 3-7. A quasi-standard C*-algebra with a minimal primal ideal which is notprime.

Let X be a topological space. Recall that a point x eX is said to be a P-point if eachfeCb(X) is constant in a neighbourhood of a;. For a non-trivial example let a>1 denotethe compact, Hausdorff space of ordinals of cardinality less than or equal to w,equipped with the order topology. It is easily seen that w is a non-isolated P-pointin o)1.

Now let X be a compact, Hausdorff space with a non-isolated P-point p. Let A bethe C*-algebra of those continuous functions/from X into the 2 x 2 complex matricesfor which f(p) is diagonal, tha t is

A =

It is easy to show that A is quasi-standard, see [3] page 357 for example. Let P bethe ideal of A consisting of those functions which vanish a tp , or equivalently, vanishin a neighbourhood of p. Then it follows from Theorem 33 and [12; 7-15] that P isa minimal primal ideal of A. However P is not prime because P = ker A ft ker/i.

48 DOUGLAS W. B. SOMERSET

4. Topologically semi-primal Banach algebras

The main purpose of this section is to show that minimal primal ideals are primein a certain class of Banach algebras. As in Section 3, the technique is to map thealgebra onto a distributive lattice in such a way that the minimal primal ideals aremapped to minimal prime ideals in the lattice. In this section the distributive latticesused are actually complete Boolean algebras.

Definition. A Banach algebra is said to be topologically semi-primal if its closedprimal ideals have zero intersection.

The class of topologically semi-primal Banach algebras is large. Since eachprimitive ideal is prime, and therefore primal, it includes all semi-simple Banachalgebras, and, in particular, all C*-algebras. Each topologically semi-primal Banachalgebra is semi-prime (because each primal ideal must contain any nilpotent ideals),but it is not known whether the converse is true or not, even in the commutativecase. A related counter-example is given in [30].

Recall that a Boolean algebra L is a distributive lattice, with a largest element 1,in which each element leL has a complement, that is an element ->/ satisfying£ V ->l = 1 and IA -i I = 0. A Boolean algebra is complete if it is complete as a lattice,that is if it has arbitrary meets and joins.

A closed subset of a topological space is said to be regular closed if it is equal tothe closure of its interior. It is well known that if X is a topological space then R(X),the collection of regular closed subsets of X, forms a complete Boolean algebra. Thelattice operations are as follows: for U, VeR(X), Uv W — U\J V, and UA V is equalto the closure of the intersection of the interiors of U and V. The complement -• U isequal to the closure of X\U.

Recall that if a is an element in a Banach algebra A then

coz (a) = {PeMin Primal (A):a$P}

and that supp (a) is a closure of coz (a) in MinPrimal(^4). Note that /£ = {beA:supp (b) ^ -> supp (a)}.

If A is a Banach algebra let R(A) denote the complete Boolean algebra of regularclosed subsets of Min Primal (̂ 4).

PROPOSITION 4-1. Let A be a topologically semi-primal Banach algebra. IfMin Primal (̂ 4) is separable then R(A) = {supp (a): aeA}.

Proof. If aeA then clearly supp (a)eR(A). Conversely, we need to show that eachregular closed subset of Min Primal (A) is equal to supp (a) for some aeA. I t is enoughto show that if F is an open subset of Min Primal (A) there exists ye A such thatcoz (y) is dense in V.

Let B = {aeA: coz (a) ci V) and set W = UaeBcoz(a). First we show that W isdense in V. For suppose that ReV and U is some basic open neighbourhood of Rcontained in the open set F, of the form U = {Pe Min Primal (A): P ^ I((l ^ i ^ n)}for some ideals I1,...,In in A. Because R is primal and does not contain any It itfollows that I1... /„ 4= {0} and hence that there exists Q e Min Primal {A) such thatQ-£l1...In. Hence QeU.Iiae(Ix... In)\Q then Q e coz (a) £ F, so Q e W. This showsthat W is dense in F.

Minimal primal ideals in Banach algebras 49Now let {Qt}(5i be a countable dense subset of W. For each i, fix yi eA with \\y(\\ = 1

such that yitQi- F ° r each A = (A1)i>1ei1, let yx be the element of A defined byyx = Si^i Aj ?/j. I t is easily seen that, for each i, the set Ki = {A e I1: yA ̂ QJ is a denseopen subset of I1. By Baire's Category Theorem (l^^Kf is non-empty. Let /x, belongto this intersection and set y = y . Then coz (y) is dense in W and hence in V asrequired. I

THEOREM 4-2. Let A be a Banach algebra. Suppose that A is topologically semi-primaland that MinPrimal(^4) is separable. Define a map <S>: A-*• R{A) by <5(a) = supp(a).Then <T> induces a homeomorphism, also called O, from MinSpec(^4) onto Spec (2?(.4)).

Proof. Let PeMinSpec(^). First we show that O(P) is an ideal of R(A).Let a,beP and let {QJ4>1 be a countable, dense subset of coz (a) U coz (6). For each

A = (Ai)i>1eZ1 define an element zxeA by

It is easily seen that for each i the set iiQ = {A e P: zx $ Qt} is a dense, open subset ofI1. By Baire's Category Theorem C\^.1K( is non-empty. Let w belong to this inter-section and set z = z^. Then coz (z) is dense in coz (a) U coz (b) so supp (2) =supp (a) V supp (6). But

by Theorem 2-3, so zeP. Hence supp (a) Vsupp(6)eO(P).Now let asP and let be A with supp (6) £ supp (a). By Theorem 2-3 (iii) there are

ideals I1,...,In in A such that P $ ^ (1 < t < n) and / ^ . . . /„ = {0}. Let

F = {e6MinPr imal ( J ) :Q$/ j ( l «S t ^ n)}.

If Q belonged to V fl coz (a) then Q would not contain Ia, contradicting the fact thatIaIx...In = {0}. Hence

F D coz (a) = V n supp (a) = 0 = F fl supp (6) = F n coz (b).

It follows that /(,/j . . . /„ = {0}. Hence P 3 /6 2 {6}, which shows that supp (6)e<D(P).This completes the proof that <t>(P) is an ideal in R(A). Note that it also shows that^ " ^ ( P ) ) = P, which implies that <J>(P) is a proper subset of R(A).

Now we show that O(P) is a prime ideal. To do this it is sufficient to show that foreach aeA exactly one of supp (a) and ->supp(a) is in <J>(P) [22; 1-2-6]. If supp(«)e<t>(P) then ^supp(a)£<I>(P) because O(P) is a proper subset of R(A). On the otherhand, if supp (a) $ <t>(P) then -isupp(a)e<P(P) (because then /„ £ P and 1% ={beA : supp (6) £->supp(a)}).

It is clear from the fact that P = O~1(d>(P)) that <J> is an injective map fromMinSpec(^4) to Spec(R(A)). We now prove that it is surjective. Let QeSpec(R(A))and define P = <t>~l(Q) = {aeA: supp (a) eQ}. Then P is an ideal of A and d>(P) = Q.Suppose that I1,...,In are ideals of A with I1) ...,/„<£ P. For each ie{ l , . . . , TO} let^e/jXP. Then supp(a{)^Q so D"_1supp(a<) =t= 0 , since Q is primal. It follows that{0} # /„ . --la ^ / j . . . /„ . HenceP is primal. If P' is a minimal prime ideal containedin P then O(P') c (D(P) = Q. But <&(P') is minimal primal, by the previous paragraph,

50 DOUGLAS W. B. SOMERSET

so <J>(P') = <1>(P) and hence P is minimal primal. This shows that O is a bijection fromMin Spec (̂ 4) onto Spec (R(A)). Finally the proof that $ is a homeomorphism isexactly the same as the proof that 0 was a homeomorphism in Theorem 3-2. I

THEOREM 4-3. Let A be a Banach algebra and suppose that A is topologically semi-primal and that Min Primal (A) is separable. Then Min Spec (̂ 4) is compact andextremally disconnected and each minimal primal ideal of A is prime.

Proof. The spectrum of a complete Boolean algebra is compact and extremallydisconnected [28; 8-2, 22-4], so it follows from Theorem 4-2 that Min Spec (A) iscompact and extremally disconnected.

Now let P be a minimal primal ideal and let a,beA\P. Then supp (a),supp(fc)^O(P), so supp (a) Asupp(.B)i£<l>(P) because O(P) is prime by Theorem 4-2.We shall show that there exists c 6 A such that supp (acb) = supp (a) A supp (b). Thenacb will not be P, which will show that P is prime, [15; 2-1].

LetV = coz (a) n coz (b) = {Pe Min Primal (A): P $ Ia,P $ /„}.

Since supp (a) A supp (b) is equal to the closure of V it is sufficient to produce ceAsuch that coz (acb) is dense in V. We begin by observing that the setW = Uxe/1 coz (axb) is dense in V. For suppose that ReV and that U is some basicopen neighbourhood of R of the form U = {Pe Min Primal (A): P $ It (1 < i < n)}for some ideals I1,...,In in A. Because R is primal and does not contain Ia,Ib or It(l ^i^n) it follows that IaIx ...InIb 4= {0} and hence that there is aP e Min Primal (A) such that P $ Ia I1... In Ib. Then PeU and it is easy to check thatthere must exist xeA such that axb$P.

Now let {QiJi^i be a countable, dense subset of W. For each i, fix xxeA with\\xt\\ = 1 such that aXfb^Pf. ForeachA = (Af)t^1el1, let cx be the element of A definedby cA = Ti^_lXixi. It is easily seen that, for each i, the set Kt = {Ael1: ax^^P^ is adense, open subset of I1. By Baire's Category Theorem (~)™_lKi is non-empty. Let ju,belong to this intersection and set c = c/t. Then coz (acb) is dense in W, hence in V, sosupp (acb) = supp (a) A supp (b) as required. I

I t is surprising, perhaps, that Min Spec (̂ 4) should be compact even when A doesnot have an identity.

If A is a separable C*-algebra then Min Primal (̂ 4) is second countable [1; 3-4], soTheorem 4-3 applies. This generalizes the fact that the space of minimal prime idealsof C(X) is compact and extremally disconnected whenever X is a separable, locallycompact, HausdorfF space, proved (for separable, completely regular spaces) in [19].An alternative proof of the compactness and extremal disconnectedness ofMin Spec (̂ 4) when A a separable C*-algebra is given in [21].

The author does not know whether Min Primal (A) is second countable when A isa separable Banach algebra, but we will now show that it is at least separable,provided that A is also topologically semi-primal.

PROPOSITION 4-4. Let A be a separable Banach algebra and let Ybea collection of closedideals of A. Then Y has a countable subcollection A such that D{/e F} = D{/6 A}.

Proof. Let J = M{7: IeY} and let {an}n ? 1 bea countable dense subset of A. For eachaeA\J there exists an IeY such that a$I. Choose an weN and an reQ such that

Minimal primal ideals in Banach algebras 51| |o-oB| | <r < i | | o+ / | | and set Xn r = {beA: \\b-aj < r}. Then aeXn r andXn r ft Iis empty. For each Xn r obtained in this way fix an ideal /„ r 6 F such that Xn r(]In r

is empty. Let A be the collection of ideals /„ r. Then A is countable, since thecollection of sets Xn r is countable, and D{7: Ie A} = J. I

Definition. It A is a Banach algebra let Prime (̂ 4) denote the space of closed, primeideals of A, equipped with the hull-kernel topology.

It is well known that when A is a separable C*-algebra Prime (̂ 4) coincides withPrim (A), the space of primitive ideals of A [27; 4-3-6].

COROLLARY 4-5. Let A be a separable Banach algebra.(i) Prime (̂ 4) is separable.

(ii) If A is topologically semi-primal then Min Primal (A) is separable.

Proof, (i) Proposition 4-4 implies that there is a countable subset A of Prime (̂ 4)with CIA = DPrime(^4). It follows from Proposition 2-2(iii) that A is dense inPrime (̂ 4) in the hull-kernel topology.

(ii) Proposition 4-4 implies that there is a countable subset A of Min Primal (A)with DA = {0}. It follows from Proposition 2-2(ii) that A is dense in MinPrimal(^4)in the lower topology. I

When A is a separable C*-algebra it is well-known that Prime (A) is secondcountable [27; 4-3-4], but it does not seem to be known whether the same is also truefor separable Banach algebras.

Corollary 4-5 shows that if A is a separable, topologically semi-primal Banachalgebra then the conclusions of Theorem 4-3 are true for A. I t is not known if theseconclusions are true for every separable Banach algebra.

The author is grateful to the United Kingdom Science and Engineering ResearchCouncil for its financial support.

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