IC/96/202

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUASI-INVARIANT MEASURESON A GROUP OF DIFFEOMORPHISMS

OF AN INFINITE-DIMENSIONAL HILBERT MANIFOLDAND ITS REPRESENTATIONS

S.V. Liidkovsky1

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

Groups of diffeomorphisms of infinite-dimensional Hilbert manifolds are defined. Their

structure is studied. Quasi-invariant measures on them, relative to dense subgroups, are

constructed. Possible applications of measures are discussed for studying unitary repre-

sentations. Moreover, irreducible unitary representations of a group of diffeomorphisms

associated with quasi-invariant measures on a Hilbert manifold are constructed.

MIRAMARE - TRIESTE

October 1996

Permanent address: Theoretical Department, Institute of General Physics, Str. Vav-ilov 38, Moscow, 117942, Russian Federation.

1

1 Introduction.

For compact Riemannian manifolds (finite-dimensional) measures on groups of diffeomor-

phisms [?], [?] were constructed, such that measures were quasi-invariant relative to

dense subgroups. Such groups are not locally compact, therefore, they can not possess

measures quasi-invariant relative to entire groups [?].

On the other hand, groups of diffeomorphisms appear naturally in partial differential equa-

tions and mathematical physics, for example, in quantum mechanics [?], [?], [?], [?].

In such theories, weighted Sobolev, Besov and Holder spaces play a very important role

[?], [?], [?], [?]. They are actively used in theories of elliptic equations on manifolds

Euclidean at infinity. Few articles were devoted to investigations of connections and cur-

vatures of groups of diffeomorphisms Difft(M) of locally compact manifolds M over R

of classes of smoothness t (see also notations below). Such groups Difft(M) are infinite-

dimensional manifolds themselves [?], [?]. They are not locally compact, hence they

cann't have quasi-invariant measures relative to the entire group, but only relative to a

subgroup G' 7 Difft(M) (see [?] ). Quasi-invariant measures produce unitary regular

representations that can be decomposed into irreducible components (see [?] and refer-

ences in [?]). Then irreducible unitary representations and quasi-invariant measures can

be used to study a manifold Difft(M) itself and M.

In [?] a possibility for construction of a differentiable measure was outlined, but it wasn't

given precisely. The approach proposed in [?] apart from [?] was applicable also to non-

flat manifolds, but uses the Lebesgue measures on Rm , where m > dimRM, dimRM is a

dimension of M over R, so it is of no use for infinite-dimensional M. Moreover, formulas

in [?], [?] cann't be transferred automatically to Hilbert manifolds M.

In the papers [?] and [?] quasi-invariant measures \i on Difft(M) for infinite-dimensional

Banach manifolds over non-Archimedean locally compact fields were constructed. The

structure of such groups and their irreducible representations was studied. Nevertheless,

cases of Dif ft(M) of infinite-dimensional M over R were not considered yet.

This paper is devoted to the construction of quasi-invariant measures JJL on Difft(M) for

infinite-dimensional Hilbert manifolds M (over the field R). The constructed measures \i

can be chosen in addition infinitely many times differentiable relative to one-parameter

subgroups < g > of dense subgroups G".

In §2 notations and definitions are given. §3 contains results about the structure of

a group of diffeomorphisms. Quasi-invariant measures on a group of diffeomorphisms are

produced in §4, §5 contains the theorem about the existence of non-measurable repre-

sentations of Difft(M) relative to \i. This develops my previous results [?], [?], [?]

for locally compact and infinite-dimensional Banach-Lie groups and Difft(M) for lo-

cally compact M (the last two cases were considered only briefly in preceding papers).

Irreducible unitary representations of a group of diffeomorphisms associated with a quasi-

invariant measure on a Hilbert manifold are described in §6. The main results of the

present paper are deduced for the first time and given below in theorems 3.4, 4.1, 5.1, 6.1,

6.17, 6.18.

2 Notations and definitions.

To avoid misunderstandings, we first present our notations and terminology.

2.1. Definition. Let U and V be open subsets in l2. We consider a space of all

infinitely many times Frechet (strongly) differentiable functions f,g:U—*V fulfilling

(i, ii) and with a finite metric ρtβ,γ(f,h) < oo, where h is some fixed smooth mapping

h : U -> V (that is of class C°°);

(i) ρβ,γt(f,g) := sup (x£U, y^x, y£U n = 0

X > / if{)9{))\\hn, « / 3 ) 7 ( / ^ ) ) EaVO, M<t, a=(a,...,

) - ^ ( x ) ) | | i 2 i 7 +Ea=(ai )...,a») ) |a| = [t] 11™^ < * >^+H+b [D»(f (x) - 9(x))-

DZ(f(v)-9(v))]\K IVn-Vn\h forn G N := {1,2,3,...}, dn,β,γt(f, g) = dn,β,γt(f,g)(x,y),

such that

(ii) ^

here x = (xj : j e N, ^ G R) G l2,γ that is ||a;||z2i7 = {E^Li(^ j 7 ) 2 } 1 / 2 < oo, oo > γ > 0,

h = l2,0 is the standard separable Hilbert space over R with the orthonormal base {en :

n G N}, URc := (x G U : ||x|| i2 > R), f(x) = (fj(x) : j G N,fj(x) G R), t > 0, [t] is

the integral part of t (the largest integer such that) [t] < t, b = {t} := t — [t], 0 < b < 1

(for b = 0 the last term in the definition of (^n^a is omitted), Dxej := d/dxj =: dj,

Dxα+γf(x) := Dxγ(Dxαf(x)), ej = (0,..., 0,1, 0,...) with 1 in the j- th place, α = (α1,..., αn),

αj G N U 0 =: N o , |Α| = α 1 + ... + αn, β G R, < x >:= min(< x >,< y >), <

x >:= (1 + II^H^)1/2, f(x) — g(x) G l2, f|A denotes a restriction of f on a subset Ac U,

na := 1α12α2...nαn for n G N.

We denote by £ ^ ( £ / , V) the completion of such metric space, E^> := f |^ i Eβj(U, V) with

the topology given by the family (p^ 7 : j G N) in the latter case. For V = l2 and

fr(u) = 0 it is the Banach space with | |/ - g\\Et,h{UM := p^tl(f,g) = P^ ) 7(/ - 9,0) that

is, the infinite-dimensional separable analog of the weighted Holder space Cp(U', R m )

(compare with [?]) for open U' C R k , k and m G N. When γ = 0 or h(U) = 0 we omit

7 or h respectively. It is evident that each cylindrical function g(Pkx) is in Eβt(U, l2) if

5- G C |( [/ ' ,R m ) , Pk:l2^ R k is the orthogonal projection, U = (Pk)-\U'), g(Pkx) :=

(g1(Pkx),...,gm(Pkx),0,0,...). The spaces Eβt(U,V) differ from ^ ( f / , F ) =: Et(U,V) for

unbounded U if β > 0.

2.2. Definition. Let M be a manifold modelled on l2 and fulfilling conditions (i-vi)

below:

(i) an atlas At(M) = [(Uj,φj) : j = 1, ...,k] is finite, k G N (or countable, k = 00),

φj : Uj —l 2 are homeomorphisms of Uj onto φj(Uj) 3 0, Uj and φj(Uj) are open in M

and l2 respectively, (4>j o 0" 1 - id) G £ ^ (<&([/* n C / ^ y for each [/; n U3,^ 0, where

^ > 0, γ > 0, id is the identity mapping id(x) = x for each x;

(ii) T M is a Riemannian vector bundle with a projection π : T M —> M and a metric ^ in

TxM induced by || * l2 with a RMZ-structure. This means that a connector K and g are

such that gc(X,Y) is constant for each C°°-curve c : I —> M, I = [0,1] C R and parallel

translation along c of X and Y G Ξ(M), Ξ(M) := ΞTM(M) is the algebra of infinitely

differentiable vector fields on M (see 3.7 in [?];

(iii) (M, g) is geodesically complete and supplied with the Levi-Civita connection and the

corresponding covariant differentiation V (see 1.1, 2.1 and 5.1 in [?]);

(iv) the charts (Uj,φj) are natural with the natural (Gaussian) coordinates with locally

convex φj(Uj) and the exponential mapping expp : Vp —> M corresponding to V, where

Vp is open in TpM for each p G M, each restriction expp|Vp is the local homeomorphism

(see §III.8 in [?], §6, 7 in [?]) such that rinj := mfxeMrinj(x) > 0, where rinj(x) is a

radius of injectivity for expx, rinj is for entire M;

(v) M is Hilbertian at infinity, that is, there exists MR C M with M \ MR =: MR equal

to finite (or countable) disjoint union of connected open components fla, a = 1, ...,p, such

that 0~1(Qa) = l2 \Ba, where Ba are closed balls in l2, each Qa is with a metric e induced

by 4>~l and the standard metric in l2. Let a metric g for M be elliptic, that is, there exists

A > 0 such that A e ^ , ξ) < gx(ξ, ξ) for each ξ G TxM and x e M, where M f i : = [ i 6 M :

DM(X,X0) < R], x0 is some fixed point in M, dM is the distance function on M induced

by g, 00 > R > 0 (see for comparison the finite-dimensional case of M in [?]);

(vi) M contains a sequence of Mk and Nk. They are supposed to be closed -E1^-

submanifolds with finite dimensions dimRM = k for Mk and codimensions codimRNk = k

for Nk, k = k(n) G N, k(n) < k(n + 1) for each n, Mk C Ml and N k D Nl for

each k < l, M = Mk U Nk, Mk C\ Nk = dMk n <9iVfc for each k such that \JkMk is

dense in M, At(M) and M are foliated in accordance with this decompositions. These

means that (α) φi,j := φi ° 4>jl\4'j(Ui H Uj) -^ l2 are of the form φi,j((xl : l G N)) =

(αi,j,k(x1, ...,xk),γi,j,k((xl : l > k))) for each n G N, k = k(n), when M is without

boundary, dM = 0. If dM =£ 0 there is the following additional condition: (β) for each

boundary component M0 of M and Ui fl M0 ^ I we have φi : Ui fl M0 - - l , where

Hl = {(xj : j G N ) | x > °}- If Ui n M0 7 0 and Uj n M0 / D we have both images in

H1 (or in Hl with l > 1), then the foliation is called transverse (tangent respectively) to

M0. Then the equivalence relation of E1^ -atlases that produces foliated M (see also [?]

for finite-dimensional Cr-manifolds) is as usually considered.

2.3. Definition. Let M and M be two manifolds as in 2.2 with a smooth mapping (for

example, an embedding) θ : M -> M, Ω and UJ > max(0,β), β G R, t G R + : = [0, oo),

oo > γ > 0, δ and δ > 7. We denote by £^' )7(M, M) a space of functions f : M —> M with

A* ^o/o^K^^n^-a-1^))), (/„-#„) e^ft^n^a-1^))),^))for each i and j . When At(M) is finite it is metrizable by a metric (z) pptJ(f, θ) :=

Y,i,jPiJ3,1(fi,j,Oi,j) with (ii) l im i ? ^ o o p^ 7 (/ |M^,6 l ) = 0. For infinite countable At(M) we

denote by Ep (M, M) the strict inductive limit str — ind — \\m{Ep (UE, M),ΠFE,Σ],

where E G Σ, Σ is the family of all finite subsets of N directed by the inclusion E < F if

E C F, UE := [JJEEUJ, (Uj,4>j) are charts ofAt(M), ΠEF : E^(UE, M) ^ E^(UF, M)

and ΠE : Ep'tJ(M,M) are uniformly continuous embeddings (isometrical for 0 < t <

00). Evidently, E^')7(M, M) is the space of functions f of the class Ep'r/ with supports

supp(f) := cl{x G M : f(x) 0} C U E ( f ) , E(f) G Σ and 0 G W C Jf7(M, M) is open

if and only if UE\W) n E^(UE, M) is open for each B e S .

Let Hom(M) be a group of homeomorphisms of M and Diffβ,γt(M) := [f G Hom(M) :

/ and / - 1 G £7g7(M, M)] be a group of homeomorhisms (diffeomorhisms for t > 1) of

class £-«7- When At(M) is finite it is metrizable with the right-invariant metric

(iii) d(f,g) :=tf(}a{g~lf,id),

where θ is the identity map for M = M, θ = id (in this case the index θ is omitted),

(3 > 0 (see also [?] for finite-dimensional M, correctness of this definition is proved in

theorem 3.1). Henceforth, we omit tilde in E.

2.4. Definition. A Riemannian metric g for M Hilbertian at infinity is called regular

Hilbertian asymptotically, if there exist δ > 0, t' > 1, / 3 / > 0 , o o > 7 / > 0 such that

(g ~ e)*(f, 0 e E^r(,(M, R) by x for each ξ G TM, ξ = (ξx : x G M), \\£x\\h < 1 for each

xeM, s u p ? e T M ) I ^ I I ^ \\(g-e)x(£,0\\E« {M>R) < S. For spaces Epr((M,N) with M = N

or N being a Banach space over R we assume that Ω > max(0,β) and (3' > max(0,β),

f > t+1, i > 7 i n 2.2, 2.4.

2.5. Definition. Let U be open in R m and V be open in R n or l2 neighbourhoods

of 0 with m and n G N. We denote by Hβ,δl,θ(U,V) the following completion relative to

a metric qβ,δl(f,g) of a family of infinitely differentiable functions f,g : U —> V with

qβ,δl(f,θ) < 00, where θ : U —V F is a smooth mapping, l is a nonnegative integer,

/? G R, 00 > «J > 0, cfctS(f,g) = (Eo<|«|<i II < x >"+H D«(/(a;) - ^(^)) |U-) 1 / 2 , i 2 :=

L2(U,F) (for F := R n or l2,δ) is the standard Hilbert space of equivalence classes of

measurable functions h : U -^ F for which exists \\h\\L2 := (/^ |/i(x)||1/iTO((ix)))1/2 < 00,

//m is the Lebesgue measure on R m . Let M and N be manifolds fulfilling 2.2(i-vi) with

finite atlases, modelled over R m for M and R n or l2 for N, θ : M ^^ N be a smooth

mapping (for example, an embedding). Then Hβ,δl,θ(M, N) is the completion of a family

of infinitely differentiable functions g,f:M^N with κβ,δt(f, θ) < 00, where Κ is a

metric κl

β,δ(f,g) = ( ^ [ ^ ( A , , ^ ) ] 2 ) 1 7 2 (see 2.3). The Hilbert spaces Hβ,δl,θ(U, F) and

Hβ,l0(TM) are called the weighted Sobolev spaces, where Hβ,δl(TM) := [f : M -> TM :

/ G Hβ,δl(M,TM), vr f (x) = x for each x E M] with θ(x) = (x,0) E TxM for each

x E M. For infinite atlases we use strict inductive limits as in 2.3. From this definition

follows that for such f and g there exists lim^^oo ^^{JWR-, 9\UR) = 0. For β = 0 or

7 = 0 we omit β or γ respectively in Diffβ,γt(M) and in i?g7. Then each topologically

conjugated space (Hlp(TM))' =: HZlp(TM) is also a Hilbert space with the standard norm

2.6. Definition. Let G be a topological group. A Radon measure \i on Af(G,fi) (or

1/ on Af(M, ν)) is called left-quasi-invariant relative to a dense subgroup G' of G, if //</,(*)

(or v<j,{*)) is equivalent to //(*) (or z/(*) respectively) for each φ G G'. Henceforth, we

assume that a quasi-invariance factor qn(4>,g) = /j,<f,(dg) //j,(dg) (or qν(φ,x)) is continuous

by (φ,g) G G' x G (or G (G' x M ) , j u : Af(G,v) -»• [0,oo), //(F) > 0 (or ν : Af(M,ν) ->

[0, 00), ν(V) > 0) for some (open) neighbourhood V C G (or C M) of the unit element

e G G (or a point x E M), /i(G) < 00 (or ν(M) < 00 and is σ-finite respectively), where

/i^E) := n(4>-lE) for each E E Af(G,/i), Af(G,/i) is the completion of Bf(G) by /i,

Bf(G) is the Borel σ-field on G [?].

Let (M,F) be a space M of measures on (G,Bf(G)) (or (M,Bf(M)) ) with values in

R and G" be a dense subgroup in G such that a topology T on M is compatible with

G", that is, // —> h (or ν —ν h ) is the homeomorphism of (M, F) onto itself for each

h E G". Let T be the topology of convergence for each E E Bf(G) (or G Bf(M))

and W be a neighbourhood of the identity e E G such that J is dense in W, where

J := [h : h E G" C\W =: W", there exists b E (-1,1) and g(b) = h with [g(c) :

c G (—1,1)] C W”], g(c1 + c2) = g(c1)g(c2), g(0) = e are one parameter subgroups,

c1,c2 G R. We assume also that for each f G W" there are g(b1), ...,g(bk) E J such

that f = g(b1)...g(bk). A measure JJL E M (or ν G M ) is called differentiable along

g(b) in a point g(c) if n(g{b)-lE) - n(E) = (b - c)(fji'(g(c);E) + α(g(b);E)) and there

exists \im.b->ca(g(b); E) = 0 and fi'(g(c);E) G R is continuous by g(c) for each E E

Bf(G), where b and c G R, /i'(g(c);E) is called the derivative (by Lagrange) along g(b)

in g(c) (analogously for ν on M). Let by induction A(*) = //^'"^(^(ci),..., (/(CJ_I); *) and

there exists \'(g(cj);E), then it is denoted fi^(g(ci), ...,g(cj); E) and is called the j-th

derivative (by Lagrange) of JJL along (g(b1),..., g(bj)) in (g(c1),..., g(cj)), where j E N.

2.7.Note. For a manifold N = ®{Mj : j G J}, Mj = M for each j , J C N, we have that

Diffβ,γt(N) is isomorphic to S <S> Diffβ,γt(M), where S is a discrete symmetric group.

Therefore, a quasi-invariant measure on Diffβ,γt(M) provides a quasi-invariant measure

Henceforward, we assume that M and Mk are connected for each k > n and some

fixed n E N. For a finite-dimensional manifold M a space Eβ,γt (M, R) (or Diffβ,γt(M))

6

is isomorphic with the usual weighted Holder space Cβt(M, R) (or Diffβt(M) correspond-

ingly).

3 Structure of groups of diffeomorphisms.

3.1. Theorem. Let G = Diffβ,γt(M) be defined as in 2.3. Then it is a separable

topological group. If At(M) is finite, G is metrizable by a left-invariant metric d.

Proof. Let at first At(M) be finite. If f and g E G then fog'1 E G due to

theorem 2.5 [?] and ch.5 in [?] about differentiation and difference quotients of composite

functions and inverse functions, since φi o cf)~l E E^s for each i and j . At first we have

d(f,id) > 0 for / ^ id in G, since there are i and j such that fitj ^ idi,j. Then

d(hf,hg) = d(g~lh~lhf,id) = d(g~lf,id) = d(f,g), hence d is left-invariant, where

f,g,h E G. Therefore, d(f~1,id) = d(id,f), in view of 2.1 and 2.3(i,ii) we have that

d(id, f) = d(f, id), hence d(f, g) = d(g, f).

It remains to verify, that the composition map (f, g) —f / g <? from G x G —> G and the

inversion map f —f f~l are continuous relative to d. Let W = [f E G : dβ,γt(f, id) < 1/2]

and f, g E W. We have fi,j o gj,l - idi,l = (fi,j o gj,l - fi,l) + (fi,l - idi,l) for corresponding

domain as an intersection of domains of fi,j o g^x and fa. Hence, using induction by

p= 1, 2,..., [t] + 1 and the Cauchy inequality we have that there are constants oo > C1 > 0,

00 > C2 > 0 such that d(f o g,id) < C1(d(f,id) + d(g,id)) and d(f-\id) < C2d(f,id),

since limra^oo[<i^/3)7(/ij-, iditj) +dt

n^tl{gjthidjti)} = 0, [t] + I and At{M) are finite, rinj > 0

and g satisfies 2.4 [?].

Indeed, in normal local coordinates x (omitting indexes (i,j) for fi,j), M 3 x = (xj :

j & N), f = (f j : C -^ R | j G N), C open in l2, using the Cauchy inequality we get:

Ei€N(l(/°g)1 - A^1)2 < 2 (EJ | ( /o g y-#V] 2 ) 1 / 2 x (EJI^-x*\r<)2)l/2 + E J K / ° g y -gi|iγ]2+12i[\gi-xi\ir]2 and'£i>j[{dj{fog)i-8})Pp]2 < a+b+ab+2(a1/2b+ab1/2)+2a1/2b1/2,

where a = E* J € N[(<9( / ° QY ~ <*})j7«7]2, b = Y.i,A(d]9l - 5))pV}2, 5\ = I for i = I and

l i = 0 f o r e a c h l ^ i , f o g = f o g ( x ) , f , g E G .

Then we can proceed by induction for finite products of Dgα(f o g)i and Dxgl, because

Dxαid(x) = 0 for |α| > 1. For f = g~l we can express recurrently {D"f~l) by (Dxξf) with

ξ i < αi for each i, since |Α| < t. Analogously, for difference quotients, since (1 + ζ)b =

1 + E~ = i (^)(m for 0 < b < 1 and 0 < |ζ| < 1, ζ G R and (1 + ζ b ) ! + bζb + ^ ( 0 with

z : R -^ R, limc^0(^(C)/C6) = 0 [?]. For countable infinite At(M) for each f,g EG there

are E(f), JB(/" 1), E(g) and S ^ " 1 ) G Σ such that supp(f) C U E ( f ) , etc., consequently,

f(supp(f)) U g~l{supp{g~1)) C U F for some F G Σ, whence t/"1 o f E G and there is

E E T, with supp{g~l o f) c f/E. If (/7 : 7 6 a) and (<jf7 : 7 G a) are two nets converging

in G to f and g respectively, so for each neighbourhood W C G there exist E E Σ and

(3 E α such that g" 1 o fγ G W, where α is a limit ordinal.

In view of the Stone-Weierstrass theorem and 2.1(i,ii) in each ET (U, V) for open U and

V in l2 are dense cylindrical polynomial functions with rational coefficients, consequently,

G is separable, since E^^U^V) is dense in Eβ,γt(U,V). Due to conditions 2.2(i-vi) for

each open submanifold V C M with V D Mk and θ > 0 every f G Diffβt(Mk) has an

e x t e n s i o n / o n t o M s u c h t h a t f G Diffβ,γt(M) w i t h f \ ( M \ M k ) n UE(f~\id) < θ.

3 . 2 . Lemma. Let M be a manifold defined in 2.2, 2.4 with submanifolds Mk and

Nk, k = k(n), n G N. Then there exist connections kV induced on Mk by V are the

Levi-Civita connections, where V is the Levi-Civita connection on M.

Proof. For each chart (Uj,φj) we have φj(Uj) C l2 and in l2 for each sequence of

subspaces R n C R n + 1 C.../2 there are induced embeddings 4>jl(Rn) n Uj ^ 4>jl(Rn+1) n

Uj °-> Uj. The Levi-Civita connection and the corresponding covariant differentiation V

for the Hilbertian manifold M induces the Levi-Civita connection V' for each submanifold

M' embedded into M, if M' is a totally geodesic submanifold. That is, for each x G M'

and X G TXM' there exists θ > 0 such that a geodesic τ = xt C M defined by the initial

condition (x,X) lies in M' for each t with |t| < θ (§5 in [?], §VII.8 in [?]). Then using

theorem 5 in §4.2 [?] and geodesic completeness of M we can choose such M' = Mk with

dimensions dim(Mk) = k G N and Mk(n) ^ M k ( n + 1 ) ^^ ...M with \JkMk dense in M.

Each manifold Mk was chosen Euclidean at infinity, since M is Hilbertian at infinity. In

view of §VII.3 in [?] and 5.2, 5.4 in [?] k(n+\)V on Mk(n + 1) induces k(n)V on Mk(n). The

latter coincides with that of induced by V on M. Here each Mk is geodesically complete,

but normal coordinates are defined in Mk in general locally as in M also, since may be

rinj(x) < oo for x G M, so that At(M) induces At(Mk) for each k = k(n), t i e N .

3.3. Lemma.Let M be a manifold fulfilling 2.2(i-vi), 2.4, At(M) be finite and TM

be its tangent bundle. Then a metric fy in Eβ,γt (M,TM) defined in 2.3 is equivalent to

the following norm

k oo

where \f\t,p,on,j{x) = \\o{x)pf{x)\Uj\\hn, \f\2

mn,ltj(x,y) =

kVXn{kVf{x)\UJ){Yl)...)Yl)\\l,

, y ) b ) ] for each n > I, ^ G [X1, ...,Xn].

Notation to lemma 3.3. There is assumed that Xj are linearly independent (in each

x G Mk, k = k(n)) vector fields on Mk corresponding to local normal coordinates such that

\\(Xi — di)\Uj\\ < i~2s, s = max(1,γ), α corresponds toY1, ...,Yl, values ofYj are taken in

each corresponding x G Mk, here \\Q\\ := supa.eA bjeA*Aj^Oybj^o ||Q(ai,..., ap; b1,..., bq)\\/

(||ai| |...| |ap | | ||&i||---||&g||) for multilinear operator Q : A<s>p®(A*)<s'q -^ B and Banach spaces

A, A* (A* denotes topologically conjugated space) that induces norms for x G M andTxM

instead of A, σ(x) := (1 + DM(X,X0)2)1/2 , dM(x,y) is the Riemannian distance function

in M induced by g and V, x0 is some fixed point in M, τ(x,y) is the tensor of parallel

transport, ΡM(X) denotes the radius of injectivity of the exponential mapping for TM and

x E M, a(x) := min(σ(x),σ(y)), raV corresponds to Mn in 3.2.

Proof. For each U <zl2 and f,gE Eβ,γt(U,TU) we have that \\f — g\\t,i3,-y = P%-y(f,g)-

Here we use the normal Gaussian coordinates (x1, ...,xn,...) on Uj as in 3.2. In view

of lemma 1.5 [?], §7.6 [?], 2.2 and 2.4 the Crystoffel symbols Γx(V,W) are in Ef>s,

where (x,V,W) -> x(V,W), β" > max(β,0), V and W E Eβ,δt(M,TM), π o Vx =

x, π o Wx = x, π : TM —> M is the canonical projection. Then df(x'(x))/dxj =

(df(x')/dx'p)(dx'p(x)/dxj), where the sum is taken over p e N, dx'p/dxj are elements

of the unitary matrices Tφa,φb := D(φb o (f)~l) for points in Ua D Ub = 0, [xj : j 6 N]

and [x/p : p G N] are the local coordinates of the charts (Ua, φa) and (Ub, φb) respectively

[?, ?], hence ?2j{dfl{x'{x))/dxj)2 = J2p(df(x')/dx'p)2. If Xj = d/dx1 then the sum by

(Y1,...,Yk) C (X1,...,Xn) in 3.3(i) corresponds to the sum by (α1,..., αn - 1) in 2.1(i),

when k = \a\ — 1. From 2.4 we have that the norm on Eβ,γt (M,TM) is equivalent to

Ptp,i{fi9)i s m c e ^4t(M) is finite and (f)j o (f)~l for each C/j fl f/j-^ 0 with CJ > max(/3,0),

where {VXY)|ψ(p) = (DYψ(p))Xψ(p) + Γψ(p)(Xψ(p),Yψ(p))) in local coordinates of a chart

(U, ψ) with p E U. Indeed, there are constants C1 > 0 and C2 > 0 such that for each n and

? 0 we have Cx EZo ll(/-y)l^ll?>/J>«>7>i < %M^9iJ)\2 < C 2 m i n ^ j E ^ ) ll(/"

7>P. s i n c e &Ua]f/2 < &U l«il) < ^ 1 / 2 ( ^ = 1 | a j |2 ) 1 / 2 for each a j G R, where

i, j = 1, ...,k < 00, because Vy/i G -E1^^^ for h G E^r( and due to properties of Γ. Then

conditions lim^^oo pJg)7(/|M^,0) = 0 and lim^^oo ||/|M^|| t )/g )7 = 0 are equivalent, where

MRc:=M\MR.

3.4. Theorem.Let M be a manifold fulfilling 2.2, 2.4 and Diffβ,γt(M) be as in 2.3

witht > 1, β > 0, γ > 0. Then

(i)for each Eβ,γt(M, TM)-vector field V its flow η

t

is a one-parameter subgroup ofDiffβ,γt(M), the curve t —η t is of class C 1, the mapping

Exp : TeDiffβ,γt(M) —> Diffβ,γt(M), V -^ η1 is continuous and defined on a neighbour-

hood of the zero section in TeDiffβ,γt(M)

(ii) TfDiffβ,γt(M) = {VE β, γt(M, TM)|π oV = f};

(iii) (V,W)= f gf{x)(Vx,Wx)ii(dx)M

is a weak Riemannian structure on a manifold Diffβ,γt(M), where n is a measure induced

on M by φj and a Gaussian measure with zero mean value on l2 produced by an injective

self-adjoint operator Q : l2 —l 2 of trace class, 0 < fi(M) < 00;

(iv) the Levi-Civita connection V induces the Levi-Civita connection

V onDiffβt(M);

(v) E : TDiffβ,γt(M) -+ Diffβ,γt(M) is defined by

EV(V) = expη(x) ° Vη on a neighbourhood of the zero section in TvDif / | (M) and is a

E^s mapping by V onto a neighbourhood Wη = Wid or] of η & Diffβ,γt(M); E is the

uniform isomorphism of uniform spaces V and W.

Proof. Let at first At(M) be finite. In view of lemma 3.3 and [?] we have that

TfEβ,γt(M, N) = \ge Eβ,γt(M,TN) : πN o g = f], where N fulfils 2.2, 2.4, πN : TN -> N

is the canonical projection. Therefore, TEβ,γt(M,N) = Eβ,γt(M,TN) = U / T / ^ ^ M , N)

and the following mapping wexp : TfEβ,γt (M, N) —> E^ (M, N), wexp(g) = exp o g gives

charts for Eβ,γt(M, N), since TN has a finite atlas of class E™x with v> / 3 > 0 , x > 7 - In

view of theorem 5 about differential equations on Banach manifolds in §4.2 [?] a vector

field V of class Eβ,γt on M defines a flow ηt of class Eβ,γt , that is dη

t/dt = V o η

t and

r]o = e. Then lightly modifying proofs of theorem 3.1 and lemmas 3.2, 3.3 in [?] we get

that ηt is a one-parameter subgroup of Diffβ,γt(M), the curve t —η t is of class C 1, the

map Sxp : TeDiffβ,γt(M) -> Diffβ,γt(M) defined by V ->• 1 is continuous.

The curves of the form t ->• ^(tV) are geodesics for V G ^Diff^M), dE{tV)/dt is the

map m - - d(exp(tV(m))/dt = j'm(t), where γm(t) is the geodesic on M, γm(0) = η(m),

7m(0) = V(m). Indeed, this follows from the existence of solutions of corresponding

differential equations in the Banach space Eβ,γt (M,TM) and then as in the proof of

theorem 9.1 [?].

From the definition of ji it follows that for each x E M there exists open neighbourhood

Y 3 x such that ji{Y) > 0 [?]. Consequently, (V,W) > 0 for each V ^ W, since V

and W are continuous vector fields and for some x & M and Y 3 x with ji(Y) > 0 we

have Vy ^ Wy for each y e Y. On the other hand sup^,^ |gf(x)(Vx, Wx)| < oo, hence

|(V, W)| < oo. From gf(x)(Vx,Wx) = gf(x)(Wx,Vx) and bilinearity of g by (Vx,Wx) it

follows that (V, W) = (W, V) and (aV, W) = (V, aW) for each a e R. For two Banach

spaces A and B we have the following uniform linear isomorhism Eβ,γt(M,A © B) =

Eβ,γt(M,A) © Eβ,γt(M,B), where © denotes the direct sum. Therefore, Eβ,γt(M,TM)

is complemented in Eβ,γt(M,T(TM)), since TM and T(TM) =: TTM are the Hilbert

manifolds of class E^x with ν > β, χ > γ > 0. Then the right multiplication αh(f) = f°h,

f -> foh is of class C°° on Diffβ,γt(M) for each h G Diffβ,γt(M). Moreover, Diffβ,γt(M)

acts on itself freely from the right, hence we have the following principal vector bundle

vf : TDiffβ,γt(M) —> Diffβ,γt(M) with the canonical projection vf.

Analogously to [?,?,?] we get the connection V = Voh on Diffβ,γt(M). Then (V^F, Z)

+ (f, V ^ i ) = JM[< VXeYe,Ze >h(x) + < Ye, VXeZe >h{x)]ii(dx) =

fM[Xeg(Ye, Ze)]h{x)ji(dx) = X(Y, Z), since Xg(Y, Z) = g(VxY, Z) + g(Y, VXZ) (Satz 3.8

in [?]) and for each right-invariant vector field V on Diffβ,γt (M) there exists a vector

10

field X on M with Vh = X o h for each h G Diffβ,γt(M), where X := X o h (see

also 3.5 in I.[?]). If V is torsion-free then V is also torsion-free. From this it follows

that the existence of E and Diffβ,γt(M) is the Banach manifold of class E^s, since exp

and M are of class E^s, αh(f) = f ° h, f —> f o h is a C°° map with the derivative

αh : E%y(M',TN) ^ Eβ,γt(M,TN) whilst fr E > 7 ( M , M ' ) , £fc(V) :

The case of infinite At(M) may be treated analogously to the proof of theorem 3.2 in

II.[?] using the strict inductive limit topology.

4 Quasi-invariant measures on a group of diffeomor-

phisms.

At first we give few preliminary definitions and results. Then we formulate the main

theorem 4.5.

4.1. Definition. Let U and V be open in l2, suppose θ : U —> V is a smooth mapping,

00 > δ > 0. We define £"} \ $(U, V) as a completion of Q relative to the family of metrics

given below [χl,r,s,δ : l,r, s G N], Q := [f : f G E™'%(U, V), there exists t i e N such that

supp(f) C U fl R n , χl,r,s,δ(f, θ) < 00 for each l, r, s], where

(i) dl,r,s,δ(f,g) := supsupp;; j(/,0)(ra!)' <€NeC/

and lim^^oo<ij)r.)S)(5(/|[7^,5f|[7^) = 0, f in prsnS are taken corresponding to [/ n Rn , that is

/It/nan : C / n R " ^ /([/) C V (see 2.1 and 2.3), pr8>n>s{hjMi,j) =

PlsihikMUinMnVyidijl^UinMn))), r = r(n) = l(n) + 2m(n), s = s(n) = γ(n) + 2m(n),

m(n) > 0 for each n, l(n) = [t] + sign{t} + [n/2] + 3,γ(n) = β+sign{t} + [n/2] + 7/2. Here

ρs,nr(f,id)(xn+1,xn+2,...) is the metric by x1, ...,xn in Er

s(UnRn, V) for f as functions by

(x1, ...,xn), ρr

s,n,δ depends on parameters (xj : j > n), we omit θ for θ = 0.

Let M fulfils 2.2, 2.4, At(M) be finite and (φj o 0" 1 - idi,j) G ^{

{y}

})X(f/ij, k) for each

C/j fl Uj 7 0, a metric g (see 2.4 ) is of class EJ ,χ , where Ui,j are (open in l2) domains

of φj o 0" 1, l'(n) > l(n) + 2, ^'(n) > γ(n) for each n, 00 > χ ^ δ. Then we can define

E$hs(M,M) as in 2.3 and G' := Di{{lγ}

},δ(M) := [f G Diff~s(M)\ (f)~] - iditj and

(FI,J - idi,j) G ^^(f/^-,/2) for each charts {Ui,φi} and {Uj,φj} with Ui D Uj ^ 0]

with the topology given by the following family of left-invariant metrics χl,r,s,δ(f,g) :=

Xi,r,s,s(g~lf,id),k

(ii) χl,r,s,δ(f,id) := Yl dl,r,s,δ(fi,j,gi,j),

gi,j(Ui,j) = 0 G l2, fi,j G l2, φi(Ui) C l2, Ui,j = Ui,j(xn+1,xn+2,...) C l2 is the domain of

fi,j by x1,...,xn for chosen (xj : j > n), ρr

s,n,δ are dependent on parameters (xj : j > n),

11

Ui,j C R n °-> l2, when (xj : j > n) are fixed and Ui,j is the region in R n by (x1, ...,xn).

For countable infinite At(M) spaces E{{γl} } s and Di\K s are defined with the help of the

strict inductive limits as in §2. Particularly, for a finite-dimensional manifold M the group

Di{{γl}},δ(Mn) is isomorphic to Diffβl(Mn) with l = l(n), γ = γ(n), where n =

4.2. Lemma. Let G" and M be as in 4.1, then G' is the topological group dense in

Proof. Let at first At(M) be finite. The minimal algebraic group gr(Q) generated by

Q is dense in G' and Diffβ,δt(M) due to the Stone-Weierstrass theorem, since Ufc Mk is

dense in M and there is l £ N such that (n!)l > nnaS for every \a\ < t and n. It remains to

verify that G' is the topological group. Let χ3(l+1),r,s,δ(f, id) < 1/2 and χ3(l+1),r,s,δ(g, id) <

1/2 then χl,r,s,δ(f o g~\id) < Cr,s,l(χ3(l+1),r,s,δ(f,id) + χ3(l+1),r,s,δ(g,id)), where Cr,s,l is a

constant depending on r,s,l and independent of f and g, since for the Bell polynomials

there are inequalities Yn(1,..., 1) < n!en for each n and Yn(F/2,..., F/(n + 1)) < (2n)!en

for Fp := F p = (n + p)p := (n +p)...(n + 2)(n + 1) (see ch.5 in [?], theorem 2.5 in [?]).

Then the case of infinite At(M) may be considered analogously to the proof of theorem

3.1.

4.3. Definition. Let {l} := {l(n) : n E N} C Z and {γ} := {γ(n) : n G N} C R

be two sequences with M and {Mk : k = k(n),n G N} be as in 3.2 and 4.1 at first

with finite At(M). Now let H{ \ S(M,TM) denotes the completion relative to the norm

defined below ||(||{z},{7},(5 of a linear span spRK over R of K := [ζ G E™>S(M,TM) :

there exists 1 < j < k such that supp(ζ) C Uj and (|t/jnMfc G HSX s{Mk,TM) for each

n G N , ||C||{z},{7},(5 < 00, H i n d o o ||C|M^||{Z},{7},(5 = 0], where

(xi : i G N) are the local coordinates in M, (x1,...,xk) correspond to that of Mk,

|| * l(n) (l ,Tl , are taken by (x1, ..,xn) and are functions by (xn+1,xn+2,...). Here

Hγ,δl(Mk|TM) := [ζ : M k ^ TM| π(ζ(x)) = x for each ^ M t , ( e Hγ,δl(Mk,TM)] (see

2-5), ||C|U(x) = (E,J ) r a< f cb7t)^(C«, M ] 2 ) 1 / 2 ; 9 7 t )^(a, ,a, ) (^) := Eo<|a|<1(»), «vo 11^x >nγ(n)+|α| Dxα(ζi,j - ^){x)\\L2{Uz]nhs)) for n > 1 analogously to ^ in 2.5 for l > 0

and γ G R, but with αn ^ 0, ||C||fc(i) := ||(|U(fc(i)) (M TMV for l < 0 we take

suP | |T | |= 1 < x > H - T W (^r i | 3-, [(„• - a.]) ! !^^,^,^,,) instead of || < x >^)+l a l ^(Ci,,- -

a,)WllL 2 (^, n ) i 2 ,), where τ G ^ ( M f c | T M ) , < x >n= (1 + E ? = 1 ( ^ ) 2 ) 1 / 2 , Ui,j,n =

Ui,j,n(xn+1,xn+2,...) is the domain of ζi,j by x1,...,xn for chosen (xj : j > n), (xj :

j G N) are the local coordinates in M, ||Cllfc(^) are the functions of x, qγl((nn)),n,δ(ζi,j,ξi,j)(x)

d e p e n d s u p o n (xj : j > n ) , H{{γl}},δ(TM) : = [ζ G H{{γl}},δ(M,TM) : vr ζ ( x ) = x f o r e a c h

:z £ M]. For infinite At(M) we define these spaces using the strict inductive limits as in

§2.

12

4.4. Lemma.Let M fulfils 2.2, 2.4, At(M) be finite Eβt(TM) be as in 2.3, H{{γl}},δ(TM)

be as in 4.3 with l(n) = [t] + [n/2] + 3 + sign{t}, γ(n) = β + [n/2] + 7/2 + sign{t}.

Then there exist constants C > 0 and Cn > 0 suc hthat | |C||E* (TM) < C||CII{l},{γ},δ for

each ζ G H{{γl}},δ(TM), ωn = k(n)2Cn, UW^w ( T M f c ) < CnU\\Hm{TMk) for k = k(n),

l'{k) = [t]+2 + szgn{t}, i(k) = β + 5/2 + sign{t}, ξ G Hγl((kk))(TMk), where sign(θ) = 1

for θ > 0, sign(θ) = - 1 for θ < 0, sign(0) = 0, {t} = t - [t] > 0.

Proof. In view of theorems in [?] and / R n < x >^ra~1 dx < 00 (for < x >n taken in

R n, x G R n ) there are embeddings Hγl((nn))(TMn) --> cfy^TMn), since 2([n/2] + 1) > n + 1 .

M o r e o v e r , t h e r e ex i s t c o n s t a n t s Cn > 0 for e a c h k = k ( n ) , t i e N s u c h t h a t HCIL^M ( K ) —

CnU\\Hm{TMk) for each ξ G Hγl((kk))(TMk). Then Dxαf(x1, ...,xn,...) - Dyαf(y1, ...,yn,...) =

Y^=o{Da'sf{v\...,vn-\xn,...)-Daf{V\...,vn,xn+\...)){ovfeH^}^

ordinates, where f(y1,...,yn-1,xn,...) = f(x1,x2, ...,xn,...) if n = 0, α = (α1,..., α m ) ,

m G N, αi G N o . Therefore, for xn < yn we have: |Dαf(y1, ...,yn~l,xn,xn+l,...)

\Dadf(y1,...,yn-1,xn,xn+1,...)/dxn- Dadf(y1,...,yn~1, zn,xn+1,...)/ dzn\hsdzn]maS <

}{}5 < ^ > l dz'a)dzn

< C'k\\f\\{l}t{l}t& x (n+ I ) " 2 , when |α| = α 1 + ... + α m < l(k), k = fc(n') > n, m < n,

where C 1 = const > 0 is independent of n and k, here s is a number of charts in At(M).

Hence H^TM) C S*>tf(TM) and H / H ^ T M ) < C| |/ | | { 1 } > { 7 } >* for each / G H^TM),

here C = C1^ E~=i n~2 < 00, since su P ; c e M E ^ i ^(^) < E ~ 1 su P ; c e M ^-(x) for # : M -»•

[0,oo), lim^oo | | / | M ^ | | E ^ 5 ( T M ) < C x lim^oo || / | M ^ | | W ) { 7 } ) 5 = 0.

The set K in 4.3 contains cylinder ζ, that is, ζ G K with supp(ζ) C Uj Pi Mn for some

1 < j < s and k = k(n), n G N. Their linear span over R is dense in Eβ,δt(TM) and

H{{γl}},δ(TM) due to the Stone-Weierstrass theorem, consequently, H{{γl}},δ(TM) is dense in

Eβ,δt(TM), since df/dxn+1 = 0 for cylinder f independent from x n + j for j > 0.

4.5. Theorem. Let M be a Hilbert manifold and g be a Riemannian metric fulfilling

2.2, 2.4 and 4.1, l G N or l = 00. Then for each Diffβ,δt(M) =: G with 0 < t < 00

and o o > / 3 > 0 ; o o > # > 0 there exists a probability measure \i on Bf(G) and a dense

subgroup G' C G and G" C G' C G such that \i is quasi-invariant relative to G' and l

times differentiable relative to G".

The proof of this theorem is divided into several parts.

4.6. Definition. Let (M,g) fulfil 2.2, 2.4 and 4.1, 1 < q < t, 0 < γ < β. For

/ G G we can define Dξζ(x) = (f(x),VM9) G Tsr(M)|f(x), where ζ(x) = (f(x),θ(x)),

θ(x) G Trs(l2), r,s G N o , ξ G Eβ,δt(TM) := [ξ G ^ ( M , T M ) | π(ξ(x)) = x] is the space

of vector fields on M of the class Eβ,δt, ζ G fEγ,δq(M,Tsr(M)) := [ζ G Eγ,δq(M,Tsr(M)) :π ( ζ ( x ) ) = f(x) for e a c h x e M], V is the covariant differentiation of tensor fields over

M, /*£ denotes the push-forward of ξ, that is, related with a pull-back /* by /* = Z" 1

13

(see 3.9(iv) in [?]), Tsr(l2) is the tensor space of type (r, s) over l2, Tsr(M) is the tensor

bundle of type (r, s) over M, TM corresponds to (r, s) = (1, 0) [?], (df)ξ := /*£ =: Dξf,

(Dζ)(ξ) := ^ K ( a O ) , V^)(X 1,...,X,) = (W)(Xi, ...,X s;ξ), X j(x) G T xM, X j G Ξ(M),

j = 1,...,s G N, r G N, ( d / ) " 1 ^ ) := ( ^ ( ( O r ) ) ] ) " 1 ^ ) G Tf-H<c{x))M; df and V"M/

are well defined for f G G analogously to §4 [?]. Indeed, the differential /* = df is a

section of T*M ® f*TM (with the induced connection in f*TM).

4.7. Lemma. In the notation of 4.6 Dξζ(x) G fE*~\tS(M,TZ(M)) and Dξ, (df)~l

are continuous mappings of fEγ,δq(M,Tsr(M)) into fE

q~ls(M,T^(M)).

Proof. Let [Mk : k = k(n),n G N] be a sequence of submanifolds as in 3.2 with

atlases At(Mk) = [(Uj,k,φj) : j] = At(M) n Mk, that is Uj,k = Uj n Mk for each j,k. If

h G Trs(l2) then \\h\\ is qiven analogously to | |Q|| in 3.2. Then fcVx,- in 3.3 corresponds to

fcV for M k in 3.2. From definitions in §2 and 4.6, lemmas 3.2 and 3.3, theorems 3.1 and

3.4 it follows that Dξ and (df)'1 are continuous, since /*£ G Et^ls(M,TM) and (df)'1

corresponds to /* = f~x, t > q > 1.

4.8. Lemma.Let Dξ be as in 4.6, f and φ G Dif fβ,δt(M) as in 4.5. Then

(4.1) £ D^^D^^of) ^

(1 )...Dξσ( l 1 )f,...,Dξσ(( i 1

/> -,^ ( i l I l +...+ i m_ l I m_1 + ( i m_1 ) I m + 1 ) ...Dξσ(l)f), where Dξf = f*Z> Dξl...Dξ2Dξ1f= V^/,&...V^/^/^i /or / > I, L>?0 o / = 0*/*^; ^ is £/ie symmetric group of {I,...,/}

elements of which are considered as bijective mappings σ : {1,...,l} —{ 1,...,l}, S is

the symmertizer of (Di1+...+imφ)(a,..., z) by all arguments (a,...,z) = (Dξσ(1)...Dξσ(l f,

•••>Div(k-im+i)---Div(i)f)> Sg(ai,...,ap) := Y,a^sp g(aσ(1), ...,aσ(p)) for a function g of ar-

guments a1,...,ap; Sw(z) denotes the summation by all partitions Ω(L) of l, that is by

all representations of l as l1i1 + ... + lmim = l, ij > 0 for each j = 1,...,m, m > 1,

h>k> ••• >lm>0.

Proof. For f and φ G Diffβt(M) and [t] > l in view of theorem 2.5 in [?], §5.1-5.3

in [?] and improving lemma 1 in [?] we have the equality (4.1) due to lemma 4.3

4.9. Definition. Let (M,g) and Dξ be as in 4.6, let us denote with the help of 4.8

the following expression

(4.2) Df,ξ1,...

-^+^CD^y.-D^J,..., Dztik_lm+iy..Dt<tmf, where < G fE«( M,Ta{M)),

[q] > l, β > γ > 0.

4.10. Lemma. Let (M,g) and {Mk : k = k(n),n G N } be as in 2.2, 2.4 and 4.1 with

each atlas At(Mk) inherited from At(M). Then there exists the locally finite partition of

unity {ψi : i G J}, J c N , for M such that

14

(i) Vi C supp(ψi) C Up(i), Vi are open, φp(i)(Vi) are locally convex, p = p(i) E {i,...,s},

U e J Vi = M;

(ii) vector fields [ ^ : i E N] are in Ξ(M) of class E\ K , supp(ξl,i) C supp(ψl), ^ E

Ξ(Mi) for each i;

(iii) [ξl,i(x) : i G N] is a linear basis in TxM for each x E Vl;

(iv) for each l E J there exists x E Vl with ξl,i(x) = ei for every i, where ei is the standard

basis in l2;

(v) 1/2 < infM \\^,i\\Et'p,iX(TVi) < S U P M Ui,i\\E*'p,iX(TVi) < 2 and s u p ^ ^ l e J > x e V l ) ; a n r f i e N

|(ξl,i(x),ξl,j(x))l2,χ| < 1/2 for some t' > 1, /?' γ i{k{l)).

Proof. The manifold (M,g) is Riemannian and modelled on l2, hence it posesses the

partition of unity {ξl : l E J}, J C N, of class E} ,χ fulfilling (i) due to §2.3 in [?], that is,

J2ieJ ψl(x) = 1 for e a c h x ^ M, ψl(x) > 0 for each l and x, supp(ψl) := cl(x E M : ipi(x) =£

0) C Up(l), cl(B) denotes the closure of B C M. Let {Mk : k = k(n),n E N} be as in 3.2

then exp for M induces exp for Mk as restrictions on corresponding neighbourhoods of

the zero sections in TMk. Therefore, the Gaussian coordinates in M induce corresponding

coordinates in Mk, since each Mk(n) has tubular neighbourhoods in Mk(n+j) (for j > 0)

and in M (§4.4-4.6 [?]). Hence for each ξ E Ξ(Mk) there is the equality | (x) = T,iejii(x),

where £i(x) = ψl(x)ξ(x). There are embeddings Ξ(Mk(n)n) ^ E(Mk(n+i)) ^ ...Ξ(M)

due to conditions of being Hilbertian at infinity for M and 2.4 on g.

Then with the help of Gaussian coordinates, the base {ej : j E N} in l2 and parallel

translation along geodesics we can choose by induction [ξi : ξi(x) E TxMi for each x E Mi

and of class E} ,χ x on M] with ξl,i = ψlξi such that to satisfy (ii-v), since Mk and M

are geodesically complete, (φj o (f)~ — id) are in the class E} ,χ for each Ui fl Uj ^ 0,

T^4>j,4>i : = -D(0i ° 0J 1 ) ' ^ j A ~ - a r e m ^ n e class E\,?^T+-^\ X, T ' ( ^ ) > /3, T1^,^ are the

unitary operators in l2 for each z in the domain of φ o 0" 1, fibers in the tangent bundle

TM are isomorphic to l2, fibers in TMi are isomorhic to R i, R i ^^ l2 (see ch. VII in [?]

and ch. I in [?]), where I : l2 —l 2 is the identity operator.

4.11. Definition. Let [fM : l,i] be the same as in 4.10 f E Diffβ,δq(M), q >

deg(An;m(n)). Let us define operators:

(4.4) An;m(n)(f(x)) = y^4>i(x)An.m(n\ f ( 6 i ) - - ) 6 n ) )

Where A ^ r n ^ A ^ i , •••,6,n) : = 2J"=1 ^s,n,m(n){gx n , •••i9xn i ^(.n 1 •") ^iis-l ' ?i 1 1 '

•••, ^ | ! n ) Ep=o «(P> 2 i ( s ) ) ^ | s [(^/)~1°-D|i( s

P/] with s := min i j>2m(ra) j , deg(An;m(n))

is a degree of An;m(n) as the differential operator, n and m(n) E N, α(p,j) E Q,

a = i(1) + ... + i(n) = 2m(n)n, gx

,

ni,j are components of g~jn on Mn, the Riemannian

metric ^ ^ on Mn is induced by gx on M for each x E Mn, (gx,n)i,j = gx,n(9/dx\ d/dxj),

Fs,n,m(n) are operators of polynomial types by gx

,

ni,j and Dξr,p.

15

4.12. Lemma. Let the operator An;m(n)(f) and f be the same as in 4.11, At(M) be

finite. Then there exists Fs,n,m(n) andα(p,j) such that An]m^{E{V)) is the continuously

Frechet differentiable by V mapping from Y®(Eq

M(TM))rMm^n into E^^l/M, TM)

for q > 4m(n)n with VvAn.m{n)\y = K2m^n + Qn : Eβ,δq(TM)

S (ra)( E(V))\f= m : fElra;m(ri)(0o/)-Ara;m(ri)(/) G

is the Beltrami-Laplace operator for Mn given in 3.2 and VLn is a differential operator for

( G E^(Mn\TM) (see definition 4.3) linear by the first argument with deg(Qn) < 4m(n)n,

(p G Diffβ,δq+2m(n)(M), supp(φ) := cl(x G M : φ(x) ^ x) C ^(^(Uj) n R n ) for some

j G {1,...,s} ,W is some open neighbourhood of id in Diffβ,δq(M), φ G W, f G W,

E(Y) = W, Y is an open neighbourhood of 0 in TidDiffβ,δt(M) (see 3.4).

Proof. For the submanifold Mn in M with the covariant differentiation raV the tor-

sion tensor Tfr = 0, nVXgx,ni,j = n^x(9x,n)i,j = 0, where X G Ξ(Mn). In view of propo-

sition III. 7.6 [?] the curvature tensor field is given by the equation Rj,k,li = (dT}j/dxk

-dYlJdxl)+Y.m^ZVl,m - r ^ H , J in local coordinates (^) , where Ei gi,^ = (dgk,t/dx^

+dgjtk/dx% —dgjti/dxk)/2, here g = gx,n (see corollary IV.2.4 [?] and 4.7), or R(X, Y,

,Zφ(p))) in local coordinates in infinite-dimensional case (§8.3 [?]). Conse-

quently, the Riemannian connection Γ in Mk and R are in the class E^ , ω > β,

X > β. For tensor fields Sj(1), ...,j(b) on Mk in the normal local coordinates we have

[V^VfcJS^i),...^) = -RP

j(1)^kSp,j(2),...,j(b) ~ ... - RP(b),r,kSj(i),-,j(b-i),P, where [Vr, Vfc] =

VrVfc- VfcVr. Then D(((J) o f)(x) = (0*)(/(x))(JD|/(^))+ £fc=! (V) [^(^X/W)]

f eS(Mra), E?:=0for j = 1.

There are constant coefficients α(j,u) fulfilling the following system of linear algebraic

e q u a t i o n s J2P=d C - d ) \ d ) a ( J > u ) = 0 w i t h d = 0 , . . . , w — 1 , w = 1, . . , m i n ( 2 m ( n ) — 1 , u ) = : p

and Ej=oQ !(i)'u) = 1, since this system is equivalent to J2P

=k (uZk)a(J>u) = 1 for k =

0, 1,...,p, where U > p, det{(^zl)}j,k 7 0, Q = 0 for a < dor d < 0, (j) := a!/(d!(a-d)!)

for d = 0,..., a are the binomial coefficients.

Using the following facts: (i) the equality [V?,Vj] = E«=o Vf[Vi, VjjVf"0"1 for p =

2, 3,..., V° := I for infinitely differentiable vector fields; and then (ii) for the correspond-

ing to V pseudo-differential operators with additional terms belonging to 5*"°° with the

well-known rules for their compositions [?]; (iii) the coefficients α(j,u) as above; (iv)

smoothness of Γ and R; (v) the expression of the Beltrami-Laplace operator in normal

coordinates for the Levi-Civita connection in Mk: K.k = {gx,k)'1'^i^j (see note 14 in v.2

[?]); (vi) lemmas 3.2, 4.3, 4.4 and 4.6 - we can find polynomials Fs,n,m(n) by Dξl,i, g with

coefficients depending on x as functions in £ I^x(Mfc,R) such that to fulfil demands of

4.12, since Bp,i(x) are polynomials of Dξjf with j = 1,..., i — 2m(n) and (Dξjφ)(f(x)) with

16

1 < j < i. The differentiability by V follows from the existence of E^Q 2:n} C Eω,

mapping of some neighbourhood Y of 0 in the Banach space TDiffβ,δq(M) onto a neigh-

bourhood W of id in Diffβ,δq(M).

Indeed, there is a neighbourhood W of id E Diffβ,δq(M) such that W2 C U and it is given

with the help of E in 3.4(v) (analogously for the class of the smoothness of M considered

here). Hence the differentiability by f E W can be reduced to the differentiability of

An;m(n) by X E Y.

4.13. Definition. Let ψ E G' (see 4.1) and An;m(n) be chosen as in 4.11 together

with {ξr,j} and constants Bn > 0. We denote A(ψ) := E~=i BnAk(n);m(k(n))(ψ)en E

®^L\(HhA (TM) ®e'n) and give below conditions when A is well defined in a neighbour-

hood of id, where {e'n : n E N} are linearly independent vectors.

4.14. Lemma.Let A and ψ be as in 4.13 and At(M) be finite. Then there are

the Banach space H, neighbourhoods of W 3 id in W C U C Diffβ,δt(M) with the

topology induced from H{{γl}},δ(TM), E(WO) =: U, W0 is some open neighbourhood of 0 in

H{{ΓL}},Δ(TM), W is open in U, V 3 0 in H and {ξr,j} such that A : W —> V is a uniform

isomorphism, where inf — linin^oo m(n)/n = c > 1 and N 3 2m(n) > n for each n.

Proof. In view of the results in [?, ?, ?, ?, ?] and lemma 4.12 A'n.m{n) : Hγl((nn)),δ(Mn|TM)

^^(n)~+Im(nyLs(-^n\TM) are the linear uniform isomorphisms for each n E N, where

ra;m(n) : = VyAra;TO(ra)(E(V))|y=0, for 4m(n)n > l(n) are considered their continuous ex-

tensions as pseudo-differential operators defined modulo terms in the class S™0. Let Bn be

such that \\BnA'k{n)Mk{n))\\ = 1 for each n. Since TG' C H{{γl}},δ(M,TM) C Eβ,δt(M,TM)

we can consider the following restriction E\H\ { γl}},δ(M,TM). Then there exists continu-

ous VyA(E(V)) = J2^Lie'n x BnVvAk(ny,m(k(n))(E(V)) in some neighbourhood of 0 in

H{{γl}},δ(M,TM) such that A' = Y.ne'nBnA>k{n)Mk{n)).

Now let H := [ξ = WnBnA>k{n)Mk{n))( : n E N}| ζ E H{{γl}},δ(TM)] be a linear space

with a norm ||f||H := J2n^n\\BnA'k{n)Mk{n))(\\'k{n), where || * \\'k{n) are defined as in

4.3 || * k(n) with l'(n) = l(n) — 4m(n)n and ^'(n) = γ(n) + 4m(n)n instead of l(n)

and γ(n) respectively for each n. There are {l"j(n) : n} and {γ"j(n) : n} such that

$ C H{~("f},s(™)- W e consider e^ as linearly independent vectors in

ej G Hffl,&(™)- Therefore, a set K := [( E E^S(TM) : there are

1 < j < s such that supp(ζ) C Uj and C|t/,-nMn G Hl^s{Mn\TM) for each n = k(p),p E

N, IICHH < oo, lim^^oo HCIM^IIH = 0] is dense both in H and H{{γl}},δ(TM), where s is

a number of charts in At(M). From the aforementioned isomorphisms and the Banach

theorem about the inverse operator it follows that A' : H{{γl}},δ(TM) —> H is the linear

uniform isomorphism and H = H is complete relative to || * H . Hence, A((y)) is contin-

uously differentiable operator in some neighbourhood W0 of 0 in H{ [ S(TM) by V E W0,

since TeG' C H{{γl}},δ(TM C TeDif fβ,δt(M) (see 3.4 and 4.12).

4.15. Lemma. Let the conditions of lemma 4.14 be satisfied. Then there exists a

17

neighbourhood P of id in G' with 2m(n) > n for each n, inf — lim™-^ m(n)/n = c > 1,

such that S^iy) : P xV —> Y is uniformly continuous (by (φ, V) E P x W0) differentiable

by V E V mapping, Y is a Banach space with an embedding operator of trace class

J :H^Y, where S^(V) := A[4>(A-1 (V))} - V.

Proof. Let Y be a Banach space of the same type as H in 4.14, but with l" (n) =

l(n)— 4m(n)n + 2m(n) andγ"(n) = 7 ( n ) + 4 m ( n ) n - 2 m ( n ) instead of/'(n) and ^'(n) for

each n. Hence, Y D H and the natural embedding J is of trace class (or nuclear, see §III.7

in [?]). Indeed, for the finite atlas At(M) and each chart (Uj, φj) we can consider linearly

independent cylinder functions xmen < x >θs /m! = f(x), where {en : n E N} C l2 is the

standard orthonormal base, xm := x1m1...xmss, m! = m1!...ms!, < x >s= (1 + Ei=i(^) 2 ) 1 / 2 >

s G N , e ( s ) = e G R . The linear span over R of such f(x) is dense in H and Y. Then

Daf(x) = enJ2(;)(D^xm/m\) (Da~? < x >f), where a = (a1,..., a8), (°) = n ( ; - ) ' s o

lim i^o o(^ + l).. .(δ + 1)δ /l! = cδ exists for each l, |cδ| < 00, linin^oo qn/n! = 0 for each

00 > q > 0, E " i E | m | > 2 m ( s ) , m K ! - m s ! ] " 1 < 00, since £ ~ l[2m(s)/s]-s < 00 and n! =

nne-nee(-n\2nn)1/2, |θ(n)| < 1/(12n) due to the Stirling formula, here m = (m1, ...ms),

m| := m1 + ... + m s, mi E Z.

Proof of theorem 4.5. Let at first At(M) be finite, t > 1, H = H and Y be as

in lemmas 4.14 and 4.15, then they are separable. For G = Diffβ,δt(M) let G' = Di{{ ζ

with 00 > ζ > 2(δ + 1) (see definition 4.1). In view of theorem I.4.4 [?] there exists a

separable Hilbert space Z over R such that θ : Z ^ H, Z is dense in H, θ is the inclusion

mapping and J o θ : Z ^^ Y is of trace class. From lemmas 4.2 and 4.4 it follows that G'

acts uniformly continuous from the left on W C U. There is a neighbourhood P of id in

G' such that PW c U.

Then conditions of theorem 26.2 in [?] are satisfied for the operators on Z induced by

I+Sφ for each φ E P and a countably additive Gaussian measure ν on Z with a correlation

operator B and a zero mean value induces a measure p, on Bf(W), p,(Q) := ν(A(Q)) for

each Q G Bf(W). The space G W = G0 is in the uniformity induced from H{{γl}},δ(M, TM),

G0 is separable, Lindelof and paracompact, consequently, there exists some locally finite

open covering {giW(i) : i E N} of G0 with gi E G' and W(i) C W, W(0) := W, so

/i(Q) := S^=o 2~JVi((fi'i~1Q) H W(i)) is countably additive and quasi-invariant relative to

G' on Bf (G0).

Since G' C G0 is dense in Dif fβ,δt(M) the measure /i induces the measure JJL such that

H(Q) = fl(Q), where Q := [x E G0 : (h1(x), ...,hs(x)) E R], Q := [x E Dif fβ,δt(M) :

(h1(x),..., h s (x)) G R], R E Bf(Rs), hi E K := [h : Diffβ,δt(M) ->• R, h are continuous]

(R is with the standard uniformity). Indeed, the minimal σ-fields over G0 and Dif fβ,δt(M)

generated by such Q and Q coincide with Bf(G0) and Bf(Diffβ,δt(M)) respectively. If

Qi^Q2 = 0 for such Qj then Qir\Q2 = 0, hence \i is additive and has σ-additive extension

on Bf(Diffβ,δt(M)), since the uniformity in G0 is stronger than in Diffβ,

18

Then diAL^r^hA^/dr = AV o η(τ; *)hA~l for h E G' and η(τ; *) G J', where V is

corresponding to η vector field, J' = Exrp(Q0) (see 3.4(1)), Q0 C F0 n TG', F 0 is a neigh-

bourhood of 0 in TDiffβ,δt(M) as in theorem 3.4(v), Lφf := φ o f in Dif fβt(M) =: G.

Practically we have constructed just above G" dense in G to fulfil definition 2.6 and then

have denoted it also G'. Indeed, without loss of generality we may suppose that G' D G",

where G" is the minimal group generated by J1 and some countable family of [gn : n E N]

C G' \ W without converging subsequences in G' such that {JnQnW = G', W is an

open neighbourhood of id in G'. Indeed, in view of theorems 26.2 [?] and 3.4(i) above

there exists /J,'(T](T;*)E) for each E E Af{G,p) and τ E (—1,1). Moreover, for each

(η(τ; *),(f>, f) E J' x G' x G there are neighbourhoods L x U' x U 3 (η(τ; *), φ, f), L,

[/' and U are open in G', G' and G respectively, such that q^{c,{r;*y,*){^,g) are uniformly

continuous by (ζ(τ; *),t/j,g) E (LD J) x U' x U (that is, there exists such quasi-invariant

measure JJL). We have analogously //^(r/i(ri; *); . . . ; ΗB(ΤB; *);£") for b e N and for each

E G Af(G,/i) and ΤI G (—1,1), ^(r^; *) G J', 9/U(f')( i;...;%;*)(V))S') i s uniformly continuous

by(Ci(r 1 ;*) , . . . ,C f c ( r 6 ;*) ,V^)e i ( l )x . . .xL(6)x[/ / x[/ , L(i) = J 'nL ' ( i ) , L'(i) are open

in G', L'(i) 3 Ci(rij*)- Indeed, ^(rji, ...,%;*) are the quasi-invariant measures due to

the Nikodym convergence theorem (but may be without condition ^b\r]i, ...,ηb;V) > 0

for each open V). The group G is not locally compact and q^m^^^ip,g) is continuous

but not uniformly on all (J')®b x G' x G.

Now let 0 < t(1) < 1 < t and {z(i) : i = 1, 2,...} be dense in M. This is possible, since

M is separable. We may define in accordance with lemma 4.14 the following subsets of

W and W(1) C Diffβ,δt(1)(M) =: G(1), W(1)C\G=: W, W(k,t(1), c; f) := [g E W(1) :

ρ(k;g,f) < c], W(k,t,c;f) := [g E W : ρ(k;g,f) < c], where oo > c > 0, k E N,

/ G W, the mappings p(k,k';g,f) := Ea ) f tsup[[cr(x)]^| V^/" 1 ^ - ^)a>6(ar)| f2>5 : j =

0,1,...,s(1),x E F(k,k')]+ Sup[[a(x)Y^[V<l\f-1 o g - id)a,b(x) - T(x,y)Vs^(tl o

g - id)a,b(y)|l2,δ]/[d(x,y)]q(1) : d(x,y) < ρ(x); (x,y) E F2(k,k') and for (x,y) exists a

chart Ui 3 x, Ui 3 y, x ^ y] are continuous on G(1) relative to ρG(1)(g,f), F(k,k') :=

[z(k),...,z(k')] for each k< > k; ρ(k;g,f) := ρ(1,k; f,g), t(1) = s(1) + q(1), 0 < s(1) G Z,

0 < q(1) < 1, cr(x) := min[σ(x), σ(y)] for a pair (x,y), ha,b = φa o h o fa1 as in §2, so

W(k + 1,t(1),c;f) C W(k,t(1),c;f) for each k G N. Therefore, n{W/(fc, t(1), c; f) : k G

N} = Bρ(G(1),f,c) n W(1), Bρ(G(1),f,c) := {g G G(1) : ρG(1)(g,f) < c}, whence the

least σ-field A generated by the family V(1) := {W(k, t(1), c; f) : c> 0,k E N, f E W}

is such that A D Bf(W(1)). Moreover, nr=i(n^Li(Un>m W(k,t(1), 1/k; fn))) = {f} for

each f G W(1) and each sequence {fn} C W converging to f G G(1).

Then we put in(W(k,t(l),c; f)) := /i(W(k, t, c; f)) for each c > 0, f G W, k E N,

whence \i\ is finitely additive, since from E(1) n L(1) = 0 in G(1) it follows E n

L = 0 in G, where E(1) and L(1) are in V(1), E and L are corresponding sets in

V := {W(k,t,c;f) : c > 0,k G N , f G W}. From the definition of // follows that

19

V>{flkLi[{XZ=iVn>mW(k,t, 1/k; fn)]]) = 0, consequently, /n is countably additive onBf(W(1)).

Each ρ(k; g, f ) is left-invariant: ρ(k; hg, hf) = ρ(k; g, f ) , hence hW(k,t, c; f ) = W(k,t, c; hf)

and hB'(f, c) = B'(hf, c) for each h,feG, where B'(f, c) := f)Zi W(k, t,c;f),c> 0.

Therefore, limc^o[^(^/(/,C)]/[/x(JB/(/,C)]=limc^o[^i(^(G(f),/,c))]/[/x1(JBp(G(f),/,c))]

= Q^h~\f) = ^(hdfi/vM) =: q^h-'J) for each fteG'and/e G, since a>o #'(/, c) =

{f}. Taking {fn} C G with lin^^oo fn = f G G(1) we have the existence of limc^0 linin^oo

fn,c))] = limc^o^n^[fi1(hBp(G(l)Jn,c))y

G(1),f,c))]/ [/x1(JB,(G(l),/,c))] =: ^ ( / T 1 , / ) . Approximating g m (h, f)

by such quotients we have that q^ih, f) is continuous by (h, f)eG'x G(1).

Indeed, for each (h, f) G G'xG(l) and θ > 0 there is a neighbourhood (h, f) G £7'x£7(l) C

G' x G(1), for each f G U(l) there are sequences {/„} C U(1) converging to / in U(1),

{/„} cUtof with [/(l)nG = [/, fc(0), ra(0) such that | [/i(/iW(fc, t, l/fc; /ra))]/[/i(W/(fc, t, 1/fc; /„))]

-\pi(hW(k,t,l/hJn))]/[ fji(W(k,t,l/hJn))\ < θ for each k > k(0), n > n(0), ( , /) G

[/' x U(1), since W(k,t, 1/k,f) are defined with the help of p(A;;*,*), G is dense in

G(1), limc^ov(W(k,t,l/k;fn)) = 0, W(k,t,c;fn)ζ = W(k,t,c;fn) for each ζ e G

satisfying ρ(k;ζ,id) = 0 and c > 0, left and right uniformities on G(1) induced by

d(f~l o g,id) =: ρ(f,g) and d(fg~l,id) =: p(f,g) give the same topology. To each

W(k, t, c; fn) corresponds the cylinder subset of Z with the help of the pseudo-differential

operator A, θ and p(k;*,fn) : W —> R, that induces the cylinder subset Z(k,t,c; fn) of

Z.

There are continuous mappings Ψk : Z —> Z generated by p(A;'; *, *), $fc',fc : (Z, E(k)) -^

(Z,E(k')) are continuous connecting mappings for each k > k', Φk,k = id, generated

by p(A;' + 1,A;;*,*), A and the correspondence between subsets of Z and Y with help

of continuous linear functionals as above, ^y = &k',k ° Ψk for k > k', where E(k) de-

notes the least σ-field generated by the family {Z(k,t,c; f) : c > 0, f G W}, Z =

Z Pi θA(W), θ : Z —H _ff = H is the embedding. Moreover, A' has continuous exten-

sion A' : H{{β0}},δ(TM) - - H}p^^?™K S(TM) and A on the corresponding neighbourhood

W C £7 := -E VFo) of id in Hom(M) (see lemma 4.14) (W is open in U); we use the op-

erator J defined above. Therefore, F = {Ψk : k G N} together with {$fc/)fc : k > A;' G N}

and {Z, £'(A;) : k G N} generate the algebra U of F-cylinder sets of Z [?].

Then for k G N and c > 0 we have the existence of l i m r ^ limra^oo[(/i(r7(r; *)B'( fn, c)) —

»(BVn,c))]/T=\imT^\pL1(V(T;*)Bp(G(l)J,c))-^

and analogously for /i^ and Hi(k), consequently, /i[ are the quasi-invariant measures rel-

ative to G' on Bf(G(1)) with analogous properties of the quasi-invariance factor for G(1)

instead of G, but with the same G' and continuity instead of uniform continuity on corre-

sponding neighbourhoods. Indeed, let Φ(Τ) = //(r/(r, *)E) for E G Bf(G), so φ(0) = 0 if

//(-E1) = 0. Using the Jordan-Hahn decomposition /j, = fi+ — \i~ we obtain a /x-integrable

function f(η(ξ; *);g) by g for each ξ G (—1,1) (that is, the logarithmic derivative of JJL

20

along η(τ; *)) such that //(//(£; *)E) = JE f(η(ξ; *); g)l^(dg) (see the case of linear spaces

in §IV.2.2 in [?]). Therefore, for each (rj(b;*),h,f) G J' x G' x G(1) and θ > 0 there

is a neighbourhood L x U' x U(1) C G' x G' x G(1), for each / G U(1) there are se-

quences {/„} C U converging to / in U(1), {fn} C U converging to /, U(l)(~\G = U, k(0),

n(0) G N,δ > 0 such that | [M W ; *)W(k, t, 1/fc; / n ) - j U ( V ( c ; *) W(k, t, 1/k; /n))]/(6-c)

- [ M M ^ *) W(fc, i, 1/fc; /„))- MMc; *)W( fc, t, 1/fc; /„))]/(& - c) | < e for each fc > fc(0),

ra>ra(0), 0 < \b-c\ < 5, (r]'(b;*),hj) G ( i n / ) x U' x U{\).

Now let At(M) be infinite. There is a sequence E1 C ...En C En+1 C ..., En G Σ

for each n G N such that Un-E-n = N and UEn are open submanifolds fulfilling the

same conditions as M. Shrinking slightly Uj if necessary we may consider that there are

natural embeddings Gn := Diffβ,δt(UEn) C Dif fβ,δt(UEn+1 C Dif fβ,δt(M) =: G such that

G = \JnGn, since G is defined with the help of the strict inductive limit and for each g G

Dif fβ,δt(M) there is E(g) G Σ with supp(g) C U E . The groups G and Gn are complete,

hence G \ Gn and Gn +1 \ G n are open in G and Gn+1 respectively [?]. Then we choose

left-quasi-invariant (l times differentiable) measures \in on Gn relative to G'n C G'n+l and

consider G' = str-ind-limn G'n such that fj,n+1\Bf(Gn) = fin, p^n+1\{G'nx Gn) = pMn and

0 < nn(Gn) < 1 — 2~n. This is possible, since Gn are the Polish spaces and hence Radonian

[?]. The minimal σ-field generated by \JnBf(Gn) coincides with Bf(G). Therefore, the

family {/in : n} generates a left-quasi-invariant (l times differentiable) measure JJL on

Bf(G) such that 0 < fi(C) = ^(C n Gi)+ E~ = 2 ^n(C n (Gn \ Gra_i)) < 1 for each

G G Bf(G), so that JJL (and JJL^ for j = 1,..., l respectively) is σ-additive (see also

proposition 1.2.4 in [?]). Indeed, for each g G G' and x G G there are n G N and

neighbourhoods G' D V 9 ^ and G D V 3 x such that g G V n G^ =: V'n and

x e F n G n = : Vn, whence p^(g,x) = p^n(g,x) for such (n,g,x), since V and Vn are open

in G'n and Gn correspondingly.

4.16. Notes. 1. In the definition of Dif fβ,δt(M) were imposed conditions β > 0 and

lim^^oo pJg(/|M£) ) = 0, so Dif fβ,δt(M) are separable and the constructed above mea-

sures // are dependent on all the natural coordinates of charts of atlas induced by E and

TDiffβ,δt(M). Indeed, there is a sequence [Rn : n e N ] of the Euclidean spaces embedded

both in Z and TDiffβ,δt(M) such that their union is dense in the latter two spaces. On

the other hand without the condition 2.3(ii) the corresponding diffeomorphisms group

Dif^(h) contains the general linear group GL(l2) over l2. The latter contains all permu-

tations of the standard orthonormal base {ej : j G N} in l2, hence GL(l2) and Difft(l2)

are not separable and not locally compact. Therefore, Dz//*(/2) permits only cylinder

measures quasi-invariant relative to a non-dense subgroup that cann't depend on all co-

ordinates, since this is the case for unseparable Hilbert spaces [?].

2. The conditions 2.2, 2.4 and 4.1 imposed on M in the particular case of dimRM =

n G N are fulfilled for a G°°-manifold Euclidean at infinity with g regularly Euclidean

21

[?]. These conditions can be weakened with the use of pseudo-differential operators on

manifolds with roughly Euclidean ends [?].

4.17. Theorem. Let n be a quasi-invariant relative to G' measure on Bf(G) with

G := Dif fβ,δt(M) as in theorem 4.5. Assume also that H := L2(G,/i,C) is the standard

Hilbert space of equivalence classes of square-integrable (by JJL) functions f : G —> C. Then

there exists a strongly continuous injective homomorphism T : G —> U(H), where U(H)

is the unitary group on H in a topology induced from a Banach space L(H —> H) of

continuous linear operators supplied with the operator norm.

Proof. Let f and h be in H, their scalar product is given by (f, h) := JG h(g)f(g)/j,(dg),

where f and h : G —> C, h denotes complex conjugated h. There exists the regu-

lar representation T : G —> U(H) defined by the following formula: T(z~l)f(g) :=

[<K*-1,0)]1/2/(*<7), where q(z,g) = fiz(dg)/fi(dg), fiz(S) = fi(zS) for each S E Bf(G),

z E G'. For each fixed z the quasi-invariance factor q(z,g) is continuous by g, hence

T(z)f(g) is measurable, if f(g) is measurable (relative to Af(G,fi) and Bf(C)). There-

fore, (T(z-l)f(g),T(z-l)h(g)) = SGh(zg)f(zg)q(z-1,g)ti(dg) = (f,h), consequently, T is

unitary. From fiz,z(dg)/n(dg) =q(z'z,g) = q(z,(z')-lg)q(z',g) = \jiz>z{dg)/nz>(dg)]\p,z>(dg)/n(dg)]

it follows that T{z')T{z) = T(z'z) and T(id) = I, T(z~l) = T~\z).

For each v > 0 and a continuous function f : G —> C with | | / | | # = 1 there is an

open neighbourhood V of id in G' (in the topology of G'), such that |q(z,g) — 1| < v for

each z £ V and each g e F for some open F in G, id E F with n{{G \ F) < v for each

z G V, where //(d#) := |/(flO|MdflO, ^ ( ^ : = Vf(zS) for e a c h S e B f ( G ) ( s e e t h e p r o o f

of theorem 4.1). Indeed, this can be done analogously for the corresponding Banach space

from which /j, was induced, f G {f1,..., fj} =: {f}, j G N.

In H continuous functions f(g) are dense, hence | JG |f(g) — f(zg)(q(z,g))1/2|2 n(dg)\ <

4v for each finite family {f} with | | / | | # = 1 and z G V = V C\ V\ where V" is an open

neighbourhood of id in G' such that ||/(5f) — f{zg)\\n < v for each z G V\ 0 < v < 1,

consequently T is strongly continuous (that is, T is continuous relative to the strong

topology on U(H) induced from L(H —> H), see its definition in [?] ). Moreover, T is

injective, since for each g ^ id there is f G C(0, G —> C) n H, such that f(id) = 0,

f(g) = 1, and | | / | | H > 0, so T(f) 7 I. In general T is not continuous relative to the norm

topology on U(H), since for each z =£ id G G' and each 1/2 > v > 0 there is f E H with

||/||.ff = 1, such that | |/ — T{z)f\\n > v, when J := supp(f) is sufficiently small with

zJn J=0.

5 Measurability of representations of a group of dif-

feomorphisms.

5.1. Theorem. Let G = Diffβ,δt(M) be as in theorem 4.5 with a quasi-invariant measure

22

\i. Then G has the family E of non-measurable characters and weakly nonmeasurable

irreducible unitary representations with the cardinality card(E) = 2c, where c := card(R).

Proof. In view of theorems 3.4(1) and 4.5 there is some symmetric open neighbour-

hood W = W~l 3 id such that fi(W) > 0 and the following family J := [g E W :

g is with infinite order gb E W for each b 6 [-1,1] C R] is such that card(J (~\ U) = c

for each U C W open in W. Then from [?] it follows that in each neighbourhood W

of id in G a family JW C J of algebraically independent elements is dense such that

card(JW H U) = c.

Then due to card(U[n!an : n E N]) = card(a) for each card(a) > Ho [?] and the

Kuratowski-Zorn lemma [?] there are algebraic automorphisms f of G satisfying the fol-

lowing conditions: (i) card(f(V) D V) = c for each V and V open in Y; (ii) card(f(V) D

V) = c for each V and V open in W (or i n W n G'), where Y C U, fi(U \ Y) <

c x min(1,m(U)), //([/) > 0, U C W, S C W, /i(W \ S) < c x min(1,m(W))/6; (iii) S,

U and Y are symmetric, S and Y are compact, intersections of each one-parameter sub-

group with S and Y correspond to subsets {b} that contain open subsets in (—1,1) C R;

(iv) the function q(g,x) = /ig(dx)//i(dx) is uniformly continuous by g E Y (~\ G', x E S

and |q(g,x) — 1| < c x min(1,m(W))/6 for such (g,x), 0 < c < min(1,m(Y))/25, where

G' is dense in G, U is fixed and open.

Indeed, there exists a family P = {(L, f)} of subgroups L C G together with their auto-

morphisms f satisfying (i,ii) for S, U and Y also fulfilling (in — iv), but with cardinals

y instead of c for c > y > tt0 := card(N). Also let each (L, f) E P fulfils: (v) if g E G \ L

and there is not any h E L and n E N with h1/n = g, then f has the extension on

gr(L U {g}), here gr(L) = L for (L, f) E P, where gr(A) is the minimal subgroup of G

containing A such that if g E gr(A) and g1/n E G for some n E N then g1/n E gr(A)

with A C G; (vi) (L, f) < (L', / ' ) if L ^ L', L C L' and the restriction f\L = f. Also

let (L1, f')R{L, f) if (vi) is fulfilled, where R is the partial order in P. The family P is

non-void, since there are (L, f) with y = Ho, that may be constructed by induction. Then

for each totally ordered P' C P, (that is, all elements of P' are comparable by relations

>, < or =) there is sup<(P /) = infR(P') = (U(L,/)eP'(-k>)) E P, hence P contains well-

ordered subsets. Therefore, by the Kuratowski-Zorn lemma there are well-ordered P1 with

inffl(P') = (G, f) E P and the family of all such f has cardinality 2c, since card(JW) = c.

Let fj,* be the outer measure for JJL (see IX.1.9 in [?]). Let us suppose, that for each open

disjoint subsets R and P in W there is the equality (vii) /j,*(q(R) n P) = 0 (where q = f

or q = f-1 for fjL*(q(W) C\W)> fi(W)/2), hence (viii) fj,*(Z n q(Z)) = )U*(g(Z)) for each

open Z in W. Then we choose Z" 1 = Z C Z2 C R C W, JU(E) > 5c> 0, gR n E = 0 for

someg EWnYnG', there is 6 e 5 c 2 n F , card(B) = c, q(b) = g,bE G', where c < 1.

From this it follows that (j,*(q(BZ)) = fj,*(q(B)q(Z)) > (1 -c)fi*(q(Z)) > (1 -c)(fi(Z) -c)

in contradiction with (vii), since q(b)Z C P for P = gi?. Therefore, /i*(q(R) (~\ R) > 0

23

and /i*(q(R) n P) > 0 for some open disjoint R and P in W.

Let T : G ^ U(H) be some non-trivial weakly measurable irreducible unitary represen-

tation or a character, where U(H) is the unitary group on some Hilbert space H. In view

of the N.Lusin theorem [?] and hc(W) = Ho (hc is the hereditary Suslin number [?]),

there exists nontrivial function z(g) = (T(f(g))y,y) that is not mesurable relative to the

complete σ-field corresponding to \i on Bf(G) and the Borel σ-field on C, where y is in

the Hilbert space H and U(H) is the unitary group with the topology induced by the

operator norm (see also [?], [?], [?], [?], [?]).

6 Irreducible unitary representations of a group of

diffeomorphisms of a Hilbert manifold.

6.1. Theorem.Let M be a Hilbert manifold fulfilling 2.2 and 2.4, G = Diffβ,γt(M) be a

group of diffeomorphisms with t > 1, β > ω and γ > 2(1 + δ). Then (for each 1 < l < 00)

there exists a quasi-invariant (and l times differentiable ) measure ν on M relative to G.

Proof. The exponential mapping exp is defined on a neighbourhood of the zero section

of the tangent bundle TM and exp is of class E^s due to 2.2. For each x G M we have

TxM=l2. Suppose F is a nuclear (of trace class) operator on l2 such that Fei = Fiei,

where ib < Fi < ic for each i, {ei : i} is the standard base in l2, 1 — γ + 2δ < b < c < — 1.

Then there exists a σ-additive Gaussian measure λ on l2 with zero mean and a correlation

operator equal F. Therefore, expx induces a σ-additive measure ν on W 3 x, where

W = expx(V), 0 G V is open in TxM, 0 < fi(V) < 00, ν(C) = n(expj~l(C)) for each

C G Bf(W). The manifold M is paracompact and Lindelof [?], GW = M, hence there

is a countable family {gj : j G N} C G, g1 = e, W1 = W and open Wj C W such that

{gjWj : j} is a locally finite covering of M with W1 = W, g1 = id. For C G Bf(M) let

ν(C) := J2jeis! p((gjlC) fl Wj)2~j (without multipliers 2~J the measure ν will be σ-finite,

but not necessarily finite).

The following mapping Yg := (expogoexp~l) on TM for each g G G satisfies conditions

of theorems 1,2 in §26 [?]. Indeed, (dg%/dxj)ijeisi in local natural coordinates (xj) is in the

class E^r^y (see 2.4). In view of these theorems and [?] the measure ν is quasi-invariant

and l times differentiable, since ((Yg)' — I)F~l^2Q is of trace class on the Hilbert space

l2 and dgt/dt = V o gt (see the proof of theorems 3.4 and 4.5 above ), where gt = η

t,

Qx = Y.3 xjjδej, x = Y.3 xjej G l2, xi G R.

6.2. Definition. 1. Let M satisfy conditions in 2.2 and 2.4. For a given atlas At(M)

we consider its refinement At'(M) = {(Uj/ipj) : j G N} of the same class E^s such that

{Uj} is a locally finite covering of M, for each f/j there is i(j) with Ui(j) D Up exp~l is

injective on f/j for some x G Uj, exp~l(U'j) is bounded in TxM=l2. Henceforward, M will

be supplied by such At'(M) and Diffβ,γt(M) will be given relative to such atlas.

24

2. Let fj, be a non-negative measure on M relative to G = Diffβ,γt(M) such that

/i(M) = oo, fj, is σ-finite and ^(f/J) < oo for each j . Then JJL is considered on Af(M,/i).

We consider X = f L e N ^ , where Mi = M for each i. Take Ei G Af(Mi,fi), put

£" = riieN-^i) which is called a unital product subset of X if it satisfies the following

conditions:

(UP S1) Y. HEi) - 1| < oo and fi(Ei) > 0 for each i;

(UPS2) Ei are mutually disjoint .

6.3. Note. In view of 6.2 the above definitions 1.1, 1.2 and lemmas 1.1, 1.2 [?] are

valuable for the case considered here (G, M, /i) for infinite-dimensional M. Henceforward,

we denote by G the connected component of id G Diffβ,γt(M) from 6.2.2. Further,

the construction of irreducible unitary representations follows schemes of [?] for finite-

dimensional M and II [?] for non-Archimedean Banach manifolds, so proofs are given

briefly with emphasis on features of the case of a Hilbert manifold M.

6.4. Let E be cofinal with E' (ERE' ) if and only if

(CF)

E be strongly cofinal with E' (E=E' ) if and only if

(SCF) there i s n e N such that ^ ( ^ A ^ ' ) = 0 for each i > n,

where EiAE^ = (Ei \ E<) U (E^ \ Ei), Σ(E) := {E1 : E'RE}.

Put UE(E') = FlieN M-^i) f° r each E' G Σ ( E ) . In view of the Kolmogorov's theorem

[?] νE has the σ-additive extension onto the minimal σ-algebra M(E) generated by Σ ( E ) .

The symmetric group of N is denoted by SQO , its subgroup of finite permutations of

N is denoted by T.^. For g G G there is gx = (gxi : i G N), where x = (xi : i G N) G X,

for σ G SQO let xσ = ( i ; J : J e N ) , ^ = xσ(i) for each i. Quite analogously to lemma 5.5 II

[?] or 1.3 [?] we have 6.5 due to supp(g) C U E ( g ) for some E(g) G Σ and n(UE^) < oo,

where U E = Ujee Uj, (Uj,ψj) are charts of At'(M).

6.5. Lemma. Let E be a unital product subset of X. Then (i) (gE)RE for each

g G G, (ii) Σ ( E ) is invariant under G and T,^.

6.6. In view of 2.6, 6.2.1 and the proof of 6.1 we may choose \i such that for each

g G G there is its neighbourhood Wg and there are constants 0 < C1 < C2 < 00 such that

(i)Cl<qll(f,z)<C2

for each x G m and f G Wg with supp(f) C UE(g). Indeed, for each Uj there exists

y G Uj such that exp~lUj is bounded in T y M. Hence for each fixed R, 00 > R > 0, for

operators Yf = U of non-linear transformations the term \det((Yf)' (x))\~1exp{Y,'iZi[2(x —

25

Yf 1{x), el)(x, el) — (x — Yf

1{x), el)2]/ Fl} is bounded (see f after (i)) for each x E l2 with

||x|| < R. For z E M\ UE(g) we have q^{f,z) = 1. Therefore, we suppose further that JJL

satisfies (i).

If S E Af{M,/i) and /i{S) < 00 we may consider measures /ik = L1 o n E'k, νk = Hk

on E'k \ S and νk = 0 on S, suppose Ln = U7=i Mi, t^Ln = <8>?=i i, Pn

: X —> Ln are

projections, ρk(x) = νk(dx)/fi(dx). Then ρk(x) = 0 for each x £ S. Using the analog of

lemma 16.1 [?] for our case we obtain the analog of lemmas 1.4, 1.6, 1.7 and theorem 1.5

[?] , since M has a countable open base {Uj : j E N there is E e Σ such that f/j C U E } .

6.7. The manifold M is Polish, hence M is Radonian [?] and for each unital product

subset E for each i there is a compact Ei C M such that /i(EiAEi) < 2~t~1 and £i C Uh(i )

for corresponding h(i) £ Σ. Since each open covering of Ei has a finite subcovering we

may choose E[ e At(M,fi) with finite number of connected components. As in §1.8 [?]

we can construct E"RE such that E"i are mutually disjoint.

6.8. Proposition. Each unital product subset E is cofinal with E° satisfying the

following conditions:

(UP3) the closure cl(Ei0) and d(\J Ej0)

are mutually disjoint and Ei0 is open for each i and infi infj.^0^1 1 E0dM(x,y) > 0,

Ei0 C Uh(i), h(i) E Σ;

(UP 4) Ei0 and E®k are connected

and simply connected, there is n E N such that for each k > n and i E N there exists

g E G with g(Ei,k0) = Bi,k being an open ball in a coordinate neighbourhood of Mk with

g|(M \ Mk) = id and m£xedMk>yeEokdM{x,y) > 0, g{E°k) = Bi}k, where B := cl(B),

E^k := Ei fl Mk. For i ^ j , Ei0 and Ej0 can be connected by an open path Pi,j such that

Proof. In view of 2.7 M and Mk are connected for each k > n and some fixed H G N .

Then using 3.1, locally finite coverings of M and Mk [?] and shrinking slightly Ei0 such

that dE® are of class E^s analogously to steps 1-4 [?] and using properties of \i we prove

this proposition. Indeed, \i is approximable from beneath by the class of compact subsets

6.9. Henceforth, Π : Σ —> U(V(Π)) denotes a unitary representation on a Hilbert

space V(Π) over C, H{^2) denotes a Hilbert space that is the completion of Ue'es(E) H|ΠE

with the scalar product < 0i,02 > = E<re£oo lE1nE2a < 0i(a^), II(a)~102(^cr~1) >v(u)

νE(dx), where H^, := L2{E'; M(E); uE\E'; V(Π)) is a Hilbert space of functions on E'

with values in V(Π), Y, := (Π; /i, E); E'RE, E is a unital product subset of X. Then we

define a representation

26

where pE(g 1\x) := (νE)g(dx)/νE(dx), (νE)g(C) := uE(g 1 G), ρE(g|x) = FlieNΡM (g;xi),

ρM(g;xi) := q^(g-l;Xi) (see §2 [?] and 5.9 [?]).

6.10. Proposition. The formula 6.9(i) determines a strongly continuous unitary rep-

resentation of G (given by 6.2 and 6.3) on the Hilbert space H(J2).

Proof. The space H(J2) is isomorphic with the completion H'(J2) of Ue'esfe) ^'YE'

with the scalar product < f1,f2 >H'-= IF < f1(x),f2(x) >V(Π) νE(dx), where fi G

H'^(i), E( G E(E), F G M(E), Fσ for σ e E M are disjoint and supp(f1(x) f2(x)) C

Uo-eSoo Fa. Here H'\E, is a space of functions / = Qu<P, where <p £ HJh' a n d W

Q n 0 : = I^o-es(-R(<T)n((r))0, (Qu((f)))(xa) = Tl(a)~l(f)(x); (ii) R(a)(f)(x) := (f)(xa); (Hi)

U(a)<P(x) := U(a)(<P(x)), \\f\\2 = fE, \\f(x)\\2

v{n)uE(dx) < oo, since E'a for a G E ^ are

disjoint for different σ. Therefore, as in 2.1 [11] we get < Ty(g)f\, f2 >=< v1,v2 >V(Π)

x riieN/(fli?(i))ni?(2) pMig^-.XiY^^dxi) for /_,- = Qu<f>j, 4>j = XBU) ®vj, w h e r e Xc is the

characteristic function of C (see also 6.6(i)).

Let us fix J G Σ and take U(J) = UjeJ Uj C M. As in the proof of theorem 5.6

(see 6.6(i)) we can find a neighbourhood W 3 id in G and 0 < c1 < c2 < oo such

that c1 < PM{g~l\y) < c2 for each y G UJ and PM(S'~1;J/) = 1 for each y £ UJ for

each g E W with supp(g) C U J . Hence for each θ > 0 there exists W 3 id such that

| < Ty(g)fi, f2 > — < f1, f2 > | < θ, consequently, due to the Banach-Steinhaus theorem

(11.6.1 [22], [26]) there exists a neighbourhood V 3 id such that \\(Ty(g) - I)f\\\ < θ and

Ty is strongly continuous.

It is interesting to note that 6.10 may be proved from the inequality: ||TW<jf)/i —

fiWu'Cp) — |v|2 JF |f1(x) ~ fi(g~l,x)pE(g~l\x)l/2\2vE(dx). Then we consider restrictions

g|Mk and properties of (Yg)' (or g o n M \ Mk) such that card{i : supp(g) n Fi,k} < Ho

for each k G N. In view of theorems 26.1,2 [?] for each sequence gn with limn gn = e and

for each θ > 0 there is m such that IF |f1(x) — fi(g~lx)pE(g~l\x)l^2\2uE(dx) < θ for all

n > m, since there is E G Σ with supp(gn) C U E for every n > m.

6.11. Let E1,...,Er be mutually disjoint open subsets of M, ^ := i^r

i=lL2(Ei))

L2(Ei) := L2(Ei;n\Ei), G1 := n[=i G|Ei, G|Ei := {g G G : supp(g) C E i}, denote by

G(Ei) the connected component of id G Diffβ,γt (Ei), also let {Ei,j : j G Ji} be the

connected components of Ei. Then G|Ei . = G(Ei,j), since for each continuous mapping

F : [0,1] —> G we have by continuity that (i) F(θ)(Ei,j) C Ei,j for each θ G [0,1] C R and

each j G Ji. Indeed, suppose J is the connected subset of [0,1] such that 0 G J and for

each θ G J is satisfied (i). If v = sup(J) < 1 then by continuity there is w > v for which

[0,w] have the same properties as J. Hence the maximal such J coincides with [0,1].

We define and consider G(E') := U"ie^G(Ei) ):= {g = (g i: 0 : gi € G(El), supp(gi) C

UE{gi\ (UieN^(^)) e Σ for each i}. Therefore, U"jeJiG(Eij) = G|Ei. Then quite analo-

gously to lemma 3 [?] and lemma 5.12 II[?] we get that the following representation L1 of

d is irreducible: (L1(g)f)(y) = EILi PM^" 1;?/*) 1/ 2 :f(g~ ly) for f G H1, g = (gi : i) G G1

27

and y = (yi : i) E 111=1 ^> since G|Ei is dense in d := GC\Y\jeJi G(Ei,j) and L1 is strongly

continuous, G|Ei C EL? eJi G(Ei,j). Indeed, in view of proposition 6.8 G|Ei is connected,

since G is connected.

Then L1 on Gi is decomposable into irreducible components, since L1 of G(Ei,j) on

L2(Ei,j) is irreducible. In view of strong continuity of L1 on the dense subgroup G|Ei it

follows that its strongly continuous extension on Gi is also unitary. Then the rest of §3.1

[?] may be transferred onto the case considered here.

Let LE,(g)f(x) = pE{g-l\x)1'2 f{g~lx) for g E G(E'), f E Hw := L2(E', M(E)\E', vE\E'),

x E E'. Then we get the following.

6.12. Lemma.Let E' E Σ ( E ) and E[ be open and connected. Then the unitary

representation LE> of G(E') on RE1 is irreducible.

6.13. Let us consider

(i) G((E')) : = {g E G| there is k = k(n),n E N and σ E E^,

such that g{E'ik) = E'a{i)>k for each i E N and g|M \ Mk = id}, where E' = [ l i e N E

(E'i C M) satisfies (UP3 - 4) and E' E Σ(E), Ei,k = Ei n Mk. In view of the foliated

structure in M this group is dense in (ii) {g E G : supp(g) C UieN^}-

6.14. Lemma. Let E' E Σ ( E ) satisfy (UP3 — 4). Then for any σ E Soo there is

n such that for each k > n there exists g E G((E')) with g(E'ik) = EL^ k for each i,

moreover, g\E[ = id\E[ if σ(i) = i.

Proof is quite analogous to that of lemma 3.4 [?], since each Mk is locally compact

and connected, also due to properties of// induced as the image of the Gaussian σ-additive

measure. On the other hand, the latter is fully characterised by its weak distribution and

is with the Radonian property (see lemma 2 and theorem 1 in §2 [?]).

6.15. Let E' be as in 6.12, H$, = L2(E', M(E)\E', uE\E'; V(Π)), H'fE, = QnH^, (see

the proof of 6.10). For each g E G((E')) there are σ E Soo and k = k(n), n G N such that

g(E'ik) = E'am for each i e N a n d g|(M\Mk) = id. Suppose f = QΠφ, φ E H^E,. If (α) φ

depends only on {x = (xi : i)|xi E E'i>k} then (TE(g)f)(x) = p E (^- 1 | x ) 1 / 2 n(a)0(^- 1 xa) .

If (β) φ depends only on {x = (xi : i)|xi E E[ \ Mk} then (T^(g)f)(x) = f(x). Then

if φ(x) = φ1(x) x φ2(x), where φ2(x) is of type (α) or (β) and φ1 : E' -^ C is also

of type analogous to (α) or (β) then T^(g)f E H'fE,. Let Gk((E')) = {g E G((E')) :

g|(M\Mk) = id}, then \JkGk((E')) is dense in G((E')). Denote Hk := {φ E H^,\<f){x)

is constant on M \ Mk}, H'k := QΠHk. In view of proposition 6.8 we have that the

following representation TE*(g)(f)(x) = pE(g~l\x)l^2H(a)(f)(g~lxa) is irreducible, where

(f) E Hk, g E Gk((£")), x E E', σ E SQO is such that g(E'ik) = E',^ k for each i (see also

lemmas 3.5 [?] and 5.15 II [?]). Then we obtain analogously to lemma 4.2 [?] the following

lemma.

6.16. Lemma. Let F = Y\i&iFi satisfy (UP3 - 4). Then there exists F' E Σ(F)

28

satisfying (UP3 — 4) and

(UPS5) M\d(\J Fi) is connected for every N > 0.

Proof . Consider Fi,k = Fi n Mk and measures \ik on Mk induced by /i on M and

the projection Pk : l2 —> R k and choose F' such that \^k{n+i){F[k,n+r)A F i , k ( n + 1 ) —

^fc(i^,fc(n)AFi,fc(n)l < 3- i- 2( f c( r a)+ 1V(i ?i) for each k = k(n) and i , n G N . Then use the-

orem 3.1 [?].

6.17. Theorem. The unitary representation Ty of G on H(J2) is irreducible.

Proof. Considering the sequences {Mk : k}, {Gk((E')) : k} and {Hk : k}, using

6.2-6.16 and strong continuity of Ty we get from the proof of theorem 4.1 [?] that Ty is

irreducible. Indeed, we may consider A := {E' : E'=E°,E' satisfies (UP3 — 4)} instead

of A in §4.3 [?].

6.18. Theorem.Suppose TV are unitary representations of G with parameters J2i =

(ΠI; fj,, E'). Then, (Ty , H(52i)), i = 1, 2 are mutually equivalent if and only if there exists

a E Soo such that IIi= aU2 and E1 e E(E2a~1), where ( aΠ)(σ) := n ( a " V a ) .

Proof. In view of 6.8 and 6.9 we may assume without loss of generality that Ei satisfies

(UP3-A, UPS5) for i = 1 and 2. Then we consider G(1) := G((E^))C\G((E^)) C G and

G ( 2 ) := n"fceNG(Cfc), where Ck are all connected components of Ei,j(1) = Ej,i(2) (with E(

here instead of F(2) in [?]). Instead of equations (5.7) we have corresponding expressions

as intersections with Mk in both sides for some k = k(n), n G N. Using the sequences

{Mk}, {Gk((E'))} and strong continuity of Ty we get the statement of theorem 6.18

analogously to §5 [?].

6.19. Note. The construction presented above of irreducible unitary representations

is valid as well for each dense subgroup G' of Diffβ,γt(M) such that the corresponding

non-negative measure λ on M is left-quasi-invariant relative to G' and satisfies 6.2 and

6.6.

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