S.V. L UDKOVSKY
Properties of quasi-invariant measures on topological groups and associated algebras
Annales mathématiques Blaise Pascal, tome 6, n
o1 (1999), p. 33-45
<http://www.numdam.org/item?id=AMBP_1999__6_1_33_0>
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Properties of quasi-invariant measures on
topological groups and associated algebras.
S.V. Ludkovsky
Mathematics
subject classification (1991 Revision) 22A10, ~3A05.
Permanent address: Theoretical
Department,
Institute of GeneralPhysics,
Str. Vavilov
38, Moscow, 117942.
Russia.Abstract
Properties
ofquasi-invariant
measures relative to densesubgroups
are considered on
topological
groups.hlainly non-locally
compactgroups are considered such as
(i)
a group ofdiffeomorphisms Di f j(t, M)
of real or non-Archimedean manifold lit in cases oflocally
compact andnonlocally
compactM,
where t is a class of
smoothness,
(ii)
a Banach-Lie group over a classical or non-Archimedean field,(iii) loop
groups of real and non-Archimedean manifolds.Recently quasi-invariant
measures on a group ofdiffeomorphisms
w ereconstructed for real
locally compact
~! in(8, 2:i~
and fornon-locally
compact real or non-Archimedean manifolds M in[10, 12, 14, 18, 20, 21].
Such groupsare also Banach manifolds or strict inductive limits of their sequences. Then
on a real and non-Archimedean
loop
groups andsemigroups
of families ofmappings
from one manifold into anotherthey
were elaborated in[13,15, 16, 17].
On real Banach-Lie groupsquasi-invaraint
measures were constructed in[3].
This article is devoted to the
investigation
ofproperties
ofquasi-invariant
measures that are
important
foranalysis
ontopological
groups and for con-struction irreducible
representations [8, 23].
Thefollowing properties
areinvestigated:
(1)
convolutions of measures andfunctions,
(2) continuity
of functions of measures,(3)
non-associativealgebras generated
with thehelp
ofquasi-invariant
measures. The theorems
given
below show that many differences appear to be betweenlocally compact
andnon-locally compact
groups. The groups considered below aresupposed
to have structure of Banach manifolds overthe
corresponding
fields.1. Definitions.
(a).
Let G be a Hausdorffseparable topological
group. A real
(or complex)
Radon measure on is called left-quasi-invariant (or right)
relative to a densesubgroup H
ofG,
if h(or h)
is
equivalent
to ~c for each h EH,
whereB f (G)
is the Borel a-field ofG, Af (G,
is itscompletion by h(A)
:_(h-1A), h(A)
:= foreach A E
Af(G, ), d (h, 9)
:=h(dg)/ (dg) Or d (h, 9)
:=h(dg)/ (dg))
denote a left
(or right) quasi-invariance
factor. We assume that the unifor-mity
7G on G is such thatG|H
C TH,(G, G)
and(H, TH)
arecomplete.
Wesuppose also that there exists an open base in e E H such that their closures in G are
compact (such pairs
exist forloop
groups and groups of diffeomor-phisms
and Banach-Liegroups).
We denoteby H) (or Mr ( G, H))
theset of
left-(
orright) quasi-invariant
measures on G relative to H with a finitenorm
~ ~
:=supA~Af(G, ) | (A)|
cn.(b).
LetLH(G,
~,C)
for1 ~
co denotes the Banach space of func- tionsf :
: G -> C such thatfh(g)
E for each h E H and~f~LpH(G, ,C) := suph~H ~fh~Lp(G, ,C)
~,where
fh(g)
:= for each g E G. For EMl(G,H)
and v EM(H)
let
(v * )(A) := k h(A)03BD(dh)
and(q*f)(g)
:=JH f(hg)q(h)v(dh)
be convolutions of measures and
functions,
whereM (H)
is the space of Radonmeasures on H with a finite norm, v e
M(H)
and q E v,C),
that is(H|q(h)|3|03BD|(dh))1/s=: ~q~Ls(H,03BD,C) oo for 1 S oo.
2. Lemma. The convolutions
* :
: M(H)
xMl(G, H) ~ Ml(G, H)
and*
: L1 (H,
v,C)
xL1H(G, , C) ~ L1H(G, , C)
.are continuous C-bilinear
mappings
Proof. It follows
immediately
from thedefinitions,
Fubini theorem and becauseg)
EL1(H
xG, v
xC).
In fact onehas,
~03BD * ~ _ ~03BD~ ~~, ~q*f~L1H(G,,C) ~ ~q~L1(H,03BD,C)
X~f~L1H(G, ,C).
3. Definition.
For
EM(G)
its involution isgiven by
thefollowing
formula: :=
(A-1),
whereb
denotescomplex conjugated
b EC,
A E
A f (G, ).
4. Lemma. Let E
Ml(G, H)
and G and H benon-locally compact
withstructures
of
Banachmanifolds.
Then*
is notequivalent
to .Proof. Let T : G -~ T G be the
tangent mapping.
Then ,u inducesquasi-
invariant measure a from an open
neighbourhood
W of the unit e E G on aneighbourhood
of the zero section V inTeG
and then it has an extensiononto the entire
TeG.
Let at firstTeG
be a Hilbert space. PutInv(g)
=g-1
then T o I nv o
T -3
=: K on V is such that there is not anyoperator
B of trace class onTeG
such thatM03BB
CB1/2TeG
andKTeG
C whereRe(1 - 8{z))
~ 0 for(Bz, z)
~ 0 and z ~TeG, 03B8(z)
is the characteristic functional ofa, M03BB
is the set of all x ETeG
such that03BBx
isequivalent
to .1(see
theorem 19.1~25J ) .
Thenusing
theorems for induced measures from aHilbert space on a Banach space
[2, 9~,
weget
the statement of lemma 4.5. Lemma. For E
Ml(G, H)
and1 ~
p oo the translation map(q, f)
~fQ{9)
is continuousfrom
H xLpH(G,
,C)
intoLpH(G, , C).
Proof. For metrizable G in view of the Lusin theorem
(2.3.5
in~5~)
anddefinitions
of T~ and TH for each E > 0 there are aneighbourhood
V ~ e in Hand
compacts Kl
and K in G such that the closureclGVK1
=:K2
iscompact
in G with
K2
cK,
the restrictionf|K2
is continuous and E, where(dg)
:=f(g) (dg).
6.
Proposition.
For aprobability
measure EM(G)
there exists anapproximate unit,
that is a sequenceof non-negative
continuousfunctions
~~ :
G ---> R such thatfG
=1 andfor
eachneighbourhood
U ~ ein G there exists
io
such thatsupp(03C8i)
C Ufor
each i >io.
Proof follows from the Radon
property of
and the existence of count-able base of
neighbourhoods
in e E G.7.
Proposition. If
: i EN)
is anapproximate
unit in H relative toa
probability
measure v E thenlimi~~ 03C8i
*f
=f
in theL1H(G,
,C)
norm, where E
Mz(G, H), f
EL1H(G,
,C)
.Proof follows from lemma III.11.18
[6]
and lemmas2,
5.8. Lemma.
Suppose
g ELqH(G,
,C)
and(gx|H)
ELq(H,
v,C) for
eachx E
G, f
ELp(H,
v,C)
with 1 p G oo,1/p+1/q
=1,
wheregx(y)
:=g(vz) for
each x and y E G. Let and v beprobability
measures, EMl(G,H).
v E
M(H).
. Thenf*g
EL1H(G,
,C)
and there exists afunction
h : G --> Csuch that
h|H
iscontinuous,
h =f*g
-a.e. on G and h aanishes at o0 onG.
Proof. In view of Fubini theorem and Holder
inequality
we have~f*g~L1H(G, ,C)
=sup G H |f(y)|
|g(z)|03BD(dy)((ys)-1dz)sup(GH|g(z)|q03BD(dy)((ys)-1dz))1/q
x(GH|f(y)|p03BD(dy)((ys)-1dz)1/p ~
~f~Lp(H,03BD,C)
X~g~LqH(G, ,C)
X03BD(H) (G).
The
equation
:=fH f(y)03C6(y)03BD(dy)
defines a continuous linear func- tional on v,C).
In view of lemma 5 the function _:h((sz)-’-)
-.w(s, x)
of two variables s and x is continuous on H x H for s, x EH,
sincethe
mapping (s, x) (sx)-1
is continuous from H x H into H.By
Fubinitheorem
(see §2.6.2
in(5~)
~G h(y)03C8(y) (dy)
=~G ~H f(y)g(yx)03C8(x)03BD(dy) (dx)
=~x
for
each 1/1
E ,C),
sinceJc Jx
o~,where
|03BD|
denotes the variation of the real-valued measure v,h(y)
:=h(y-1).
Here 03C8
isarbitrary
inLp(G,
,C),
from this itfollows,
that({y
:h(y) ~ (f*g)(y),
y EG})
=0,
since h and(f*g)
are-measurable
functions due to Fubini theorem and thecontinuity
of thecomposition
and the inversion in atopological
group. In view of Lusin theorem(see §2.3.5
in(5))
for each e > 0there are
compact
subsets C C H and D c G and functionsf’
E v, C~and
g’
ELX(G, C)
with closedsupports supp( f’)
CC, supp(g’)
c D suchthat
clGCD
iscompact
inG,
~f’
-f~Lp(H,03BD,C)
E and~g’
- ~,since
by
thesupposition
of§1
the group H has the baseBH
of itstopology
TH, such that the closures
clGV
arecompact
in G for each V ~BH.
Fromthe
inequality
|h’(x) - h(x)|
_(~f~Lp(H,03BD,C)
+~)~
+~~g~LqH(G, ,C)
it follows that for each b > 0 there exists a
compact
subset K C G with 6 for each x E GB K,
where :=9.
Proposition.
LetA, B
E ~ and v beprobability
measures,E
Ml(G, H),
v EM(H).
Then thefunction 03B6(x)
:=(A
flxB)
is con-tinuous on Hand
03BD(yB-1
flH)
E v,C).
.Moreover, if (A) (B)
>0,
EL({y
E G: yB-1~H
EAf(H, v)
and03BD(yB-1~H)
>0})
>0, then 03B6(x) ~
0on H.
Proof. Let
gx(y)
:=chA(y)chB(x-ly),
theng2(y)
ELqH(G, , C)
for1 q ~, where
chA(y)
is the characteristic function of A. In view ofpropositions
6 and 7 there existslimi~~ 03C8i
* gz = gx inL1H(G,
,C).
Inview of lemma III.11.18
[6]
and lemma8,
is continous. There is thefollowing inequality:
1 >
rH (A
flxB)v(dx)
=fx Jc chA(y)chB(x-1y) dy)03BD(dx).
In view of Fubini theorem there exists
f H chB(x-1y)03BD(dy)
=v((yB-1)
nH)
EC),
henceJx (A
flxB)v(dx)
=fG 03BD(yB-1
flH)chA(y) (dy).
10.
Corollary.
LetA,
B EAf(G,
v EM(H)
and u EMl(G, H)
beprobability
measures. Thendenoting IntHV
the interiorof
a subset Vof
Hwith
respect
to TH, one has(i) IntH(AB)
n H~ 0,
when({y
E G: v(yB
nH)
>0})
>0;
(ii) IntH(AA-1)
e, when({y
E G :03BD(yA-1 n H)
>o})
> 0.Proof. AB n H D
f ~
E H : n >0}.
11.
Corollary.
Let G = H.If
~. E is aprobability
measure, then G is alocally compact topological
group.Proof. Let us take v = ~c and A = C U where C is a compact subset of G with
(C)
>0,
whence(yA)
> 0 for each y E G andinevitably
IntG(AA-1)
e.12. Lemma. Let tc E
Ml(G, H)
be aprobability
measure and G benon-locally compact.
Then(H)
= 0.Proof. This follows from theorem 19.2
[25]
and theorem 3.21 and lemma 3.26[19]
and theproof
of lemma4,
since theembedding TeG
is a compactoperator
in the non-Archimedean case and of trace class in the realcase
(see
also the papers about construction ofquasi-invariant
measures onthe groups considered here
[3, 8, 10; 12. 13, 14, 16, 17, 18], [20, 21, 24]).
Indeed,
the measure ~c on G is inducedby
thecorresponding
measure v on aBanach space Z for which there exists a local
diffeomorphism
A : W --~V,
where W is a
neighbourhood
of e in G and V is aneighbourhood
of 0 in Z.The measure ,u on G is
quasi-invariant
relative to H.Therefore,
the measurev on U is
quasi-invariant
relative to the action ofelements ~
E W’ C W n Hdue to the local
diffeomorphism A,
thatis,
v~ isequivalent
to v foreach §
:=where
AW’A-1U
Cv,
W’ is an openneighbourhood
of e in Hand U is an openneighbourhood
of 0 inZ, vo(E)
:=~
is an operatoron Z such that it may be non-linear. The
quasi-invariance
factorv)
hasexpression through
and thequasi-invariance
factorq03BD(z,x)
relativeto linear shifts z E Z’
given by
theorems from§26 [25]
in the real case andtheorem 3.28
[19]
in the non-Archimedean case:~det~~’(~~1~~)~?~q~(x - ~-I(x), x),
where x E
U, 03C6
= E W’. Then E Z’ for each v E Vand 03C8
EW’,
where v on Z isquasi-invariant
relative to shifts on vectorsz E Z’ and there exists a compact operator in the non-Archimedean case
and an
operator
of trace class in the real case ofembedding
0 : Z’ ~ Z suchthat
v(Z’)
= 0.13, Theorem. Let
(G, ’To)
and(H, TH)
be apair of topological non-locally compact
groupsG,
H(Banach-Lie,
Frechet-Lie or groupsof diffeomorphisms
or
loop groups)
withuniformities
TG, TH such that H is dense in(G, TG)
andthere is a
probability
measure EMl(G, H)
with continuousd (z, g)
onH x G. . Also let X be a Hilbert space over C and
U(X)
be theunitary
group.Then
(1) if T
: G -->U(X)
is aweakly
continuousrepresentation,
then thereexists T’ : G ~
U(X) equal
-a.e. to T andT’|(H,H)
isstrongly continuous;
(2) if
T : G --~U(X)
is aweakly
measurablerepresentation
and X isseparable,
then there exists T’ : G ~U(X) equal
to T -a.e. andT’|(H,H)
isstrongly
continuous.Proof. Let
R(G) :_ (I)
where I is the unitoperator
onL~.
Then we can define
’- +
G
>where
ah(g)
:= Then~) ~ _ G 9)
-a(9} I I (T9~~
>hence
Aah
isstrongly
continuous withrespect
to h ~H,
thatis,
lim |Aah03BE -Aa03BE|
= 0.h~e
Denote
Aah
=T’hAa
as in§29 [22],
soT’h03BE
=Aah03BE, where 03BE
=Aa03BE0,
a EL1.
Whence
_
(Aah03BE0, Aah03BE0) =
Gah(g)(Tg03BE0, Tg’03BE0)d (h-1, g)d (h-1,g’(ah(g’) (dg) (dg’)
-
Ga(z)a(z’)(Uz03BE0, Uz’03BE0) (dz) (dz’)
=(03BE, 03BE).
Therefore, Th
isuniquely
extended to aunitary operator
in the Hilbert space X’ C X . In view of lemma12, ~c(H}
= 0. Hence T’ may be consideredequal
to T Then the space
spanC[Aah :
h EH]
isevidently
dense inX since
(Aah03BE1, Axg03BE0)
= (~Gah(g)Tgd(h-1,g)(dg)03BE1,Gxq(g’)Tg’d(q-1,g’)(dg’)03BE0)
=(Th G
-’(Tq-1hAa03BE1, AxSO).
For
proving
the second statement letR :=
[03BE : Aa03BE
= 0 for each a EL1(G, )].
IfG ~) (dg)
=G (T’g03BE, ~) (dg)
for each
a(g)
EL1(G,
,C),
then(Tg03BE, ~)
=(T’g03BE, ~)
for-almost
all g E G.Suppose
that n EN}
is acomplete
orthonormal system in X.If 03BE ~ X,
then
= 0
for
each g
EG B Sm,
where = 0.Therefore, 03BEm)
= 0 for eachm E N, if g
EGB S,
where S :=Sm.
HenceTg~
= 0 foreach g
EGB S, consequently, 03BE
= 0. Then(Tg03BEn,03BEm)
= for each g EG B
03B3n,m, where = 0. Hence03BEm) = (T’g03BEn,03BEm)
for each n, m E N andeach g
E GB 03B3, where 03B3
:=~n,m
03B3n,m and(03B3)
= 0.Therefeore,
R = 0.14. Definition and note. Let
{Gi : i
ENo}
be a sequence of topo-logical
groups such that G =Go; Gi+1
CGi
andGi+1
is dense inGi
foreach i E
No
and theirtopologies
are denoted Ti, C T;+1 for eachi,
whereNo
:=~0,1, 2, ...}. Suppose
that these groups aresupplied
with realprobability quasi-invariant measures i
onGi
relative to Forexample,
such sequences exist for groups of
diffeomorphisms
orloop
groups considered inprevious
papers[10, 12, 13, 15, 16, 17, 18], [20, 21].
LetL2Gi+1 (G:. i, C)
denotes a
subspace
ofL2(G~, ~c=, C)
as in~1 (b}.
Such spaces are Banach and not Hilbert ingeneral.
LetL2( Gi, C))
:=H1
denotes thesubspace
ofL2(Gi, C)
of elementsf
such that~f~2i
:=[~f~2L2(Gi, i,C)
+~f~’2i]/2
oo, where~f~’2i
’-Gi+1Gi
|f(y-1x)|2 i(dx) i+1(dy).
Evidently H1
are Hilbert spaces due to theparallelogram identity.
Letfi+1 * fi(x) := Gi+1 fi+1(y)fi(y-1)x)i+1(dy) denotes the convolution of
f i
EH;.
15. Lemma. The convolution * : x
Hi
---~Hi is
a continuous bilinearmapping.
Proof. In view of Fubini theorem and
Cauchy inequality:
16. Definition. Let
12((H; :
: I eNo})
=: H be the Hilbert spaceconsisting
of elementsf
=( f" : fi
eH;,
I eNo)
for whichoo
~f~2
:=03A3~fi~2i
~.I=0
For elements
f
and g e H their convolution is definedby
the formula:f*g
=h with
h’
:= *g’
for each I eNo.
Let * : H --+ H be an involutionsuch that
f *
:=(fj^ : j
ENo),
where :=fj(y-1j)
for each yj EGj, f
:=No), z
denotes thecomplex conjugated z
E C.17. Lemma. H is a non-associative non-commutative Hilbert
algebra
with
involution *,
that is * isconjugate-linear
andf**
=f for
eachf
E H.Proof. In view of Lemma 15 the convolution h =
f * g
in the Hilbert space H has the norm~~ f (I ~~9~~,
hence is a continuousmapping
fromH x H into H. From its definition it follows that the convolution is bilinear.
It is non-associative as follows from the
computation
of i-th terms of( f *g)*q
and
f * (g
*q),
which are( f ‘+2 * gi+1)
*q=
andfi+1
*(g‘+z
*qi) respectively,
where
f , g
and q E H. It isnon-commutative,
since there aref and g
E Hfor which are not
equal
to *Ii.
Since = andZ = z. one has
f **
=( f * ) *
=f
.18. Note. In
general ( f
*g*)* #
g *f *
forf
and g EH,
since thereexist
f ? and gj
such thatgj+1 * (fj)* ~ (fj+1
*(gj)*)*.
Iff
E H is such thatf ~
=f ~+l,
then((fj-1)* * fj)(e)
=Gj+1fj+1(y-1)fj+1(y)j+1(dy)
=~fj+1~2L2(Gj+1, j+1,C),
where j
ENo.
19. Definition. Let
l2(C)
be the standard Hilbert space over thefield
C be considered as a Hilbert
algebra
with the convolution a *{3
= ~y suchthat ;’ :=
where a := EC,i
ENo),
a,/3 and,
E12(C).
20. Note. The
algebra l2(C)
has two-sided idealsJi
:={a
E12(C)
:aj
= 0 foreach j
>i},
where i ENo.
Thatis, J*12(C)
c J and12(C)*J
= Jand J is the C-linear
subspace
ofl~(C),
but J *l2(C) ~
J. There are alsoright ideals,
which are not left ideals:Ki
:={a
El2(C) : a~
= 0 for eachj
=0...., i}, where j
ENo.
Thatis, l2(C) * Ki
=Ki,
butKa * l2(C)
=K~_1
for each i E
No;
whereK-l
:=l2(C).
Thealgebra l~(C)
is theparticular
case of
H,
whenGj
=~e}
foreach j
ENo.
We consider further H for non-trivialtopological
groups outlined above.21. Theorem.
If
F is a maximal properleft
orright
ideal inH,
thenH/F
isisomorphic
as the nonassociative noncommutativealgebra
over Cwith
I~(C).
Proof. Since F is the
ideal,
it is the C-linearsubspace
of H.Suppose,
that there
exists j
~ENo
such thatf’
= 0 for eachf
EF,
then/’
s = 0for each i E
No,
since the space of boundedcomplex-valued
continuousfunctions
C)
onG~
:=G~
is dense in eachH~
:=: f
EH}
and
C) ~ Fj
={0}
andC0b(Gj,C)|Gj+1
~ C0b(Gj+1,C). Therefore, .
"{0}
foreach j
ENo, consequently,
C ~---~F~
foreach j
ENo.
Since Cis embeddable into each
Fy,
then there exists theembedding
oft2(C)
intoF,
wherejHj
:={/~ : : f
E~}, ~ :
H --~H~
are the naturalprojections.
The
sub algebra
F is closed inH,
since H is atopological algebra
and Fis a maximal proper
subalgebra.
The spaceH~
:=~j~No H; is
dense in eachHa
and the groupGoo
:=~j~No Gj
is dense in eachGj.
Suppose that Fi
=Hi
for some i ENo, then Fj
=Hj
foreach j
ENo,
since
C0b(G~, C)
is dense in eachHj
andC0b(Gj, C)|Gj+1 ~ C0b(Gj+1, C).
Theideal F is proper,
consequently, JE~
as the C-linearsubspace
for eachj
ENo, where Fj
=03C0j(F).
There are linear continuous
operators
from12(C)
into12(C) given by
thefollowing
formulas: ~ ’-~(o, ..., 0, x°, xl, x2, ...)
with 0 as n coordinates at thebeginning, x
~ for n EN;
x H k = E(~, l, ..., t-1)),
where N ~ l >2,
~~ ESl
are elements of thesymmetric
groupSi
of the set(0,1, ..., l -1).
Thenf * (g * h)
+l2(C)
and( f * g) * ~
+12(C)
are considered as the same
class,
alsof * g
+l2(C)
- g ~f
+12(C)
inHj12(C),
since( f
+12(C) ) * (9
+12(C) )
=f * 9
+12(C)
for eachf, g
andhE H. For each
f , g, h
E F:are considered as the same
class,
alsof *
g +l2(C)
= 9 *f
+~2(C)
inF//2(C),
since( f
+LZ(C)) * (g
+t2(C))
=f *
9 +l2(C)
C F for eachf
andg E F.
Therefore,
thequotient algebras
andF/l2(C)
are associative commutative Banachalgebras.
Let us
adjoint
a unit to and As a consequence of the Gelfand and Mazur theorem wehave,
that(H/l2(C))/(F’/t2(C))
is isomor-phic
with C(see
theorem V.6.12[6]
and theorem III.11.1[22]).
On the otherhand,
as it wasproved
aboveF~ ~ H~
foreach j
ENo,
hence there exists thefollowing embedding l2(C) (H/F)
and(H/F)/l2(C)
isisomorphic
with(H/l2(C))/(F/l2(C)). Therefore, H~F
isisomorphic
witht2(C).
References
[1]
N. Bourbaki. Lie groups andalgebras (Moscow: Mir, 1976).
[2]
Yu.L.Dalecky,
S.V. Fomin. Measures and differentialequations
in infinite-dimensional space(Kluwer: Dordrecht,
TheNetherlands, 1991).
[3]
Yu.L.Daletskii,
Ya.I. Shnaiderman. Diffusion andquasi-invariant
mea-sures on infinite-dimensional Lie groups. Funct. Anal. and its
Applica-
tions. 3
(1969), 156-158.
[4]
R.Engelking.
Generaltopology (Moscow: Mir,1986).
[5]
H.Federer. Geometric measuretheory(Berlin:Springer-Verlag, 1969).
[6]
J.M.G.Fell,
R.S. Doran.Representations of *-algebras. locally compact
groups, and Banach
*-algebraic
bundles(Acad.
Pr.:Boston, 1988).
[7]
E. Hewitt and K.A. Ross. Abstract harmenicanalysis.
Second edition(Berlin: Springer-Verlag, 1979).
[8]
A.V.Kosyak.
Irreducible Gaussianrepresentations
of the group of the interval and circlediffeomorphisms.
J. Funct.Anal. 125(1994),
493-547.[9]
H.-H. Kuo. Gaussian measures in Banach spaces(Springer, Berlin, 1975).
[10]
S.V.Ludkovsky.
Measures on groups ofdiffeomorphisms
of non-Archimedean Banach
manifolds, Usp.
Mat. Nauk.51(1996),
169-170(N” 2).
[11]
S.V.Ludkovsky. Measurability
ofrepesentations
oflocally compact
groups. Math.
Sb. 186 1995,
83-92(N” 2).
[12]
S.V.Ludkovsky.
Measures on groups ofdiffeomorphisms
of non-Archimedean
manifolds, representations
of groups and theirapplica-
tions. Theoret. i Mathem.
Phys., 1999.
[13]
S.V.Ludkovsky. Quasi-invariant
measures on non-Archimedean semi- groups ofloops. Usp.
Mat.Nauk,
53(1998), 203-204 (N” 3).
[14]
S.V.Ludkovsky. Quasi-invariant
measures on a groupof diffeomorphisms
of an infinite-dimensional real manifold and induced irreducible
unitary representations.
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. Nuova Serie. 26 pages, isaccepted
forpublication,
1999.[15]
S.V.Ludkovsky. Quasi-invariant
measures onloop
groups of Riemann manifolds. Dokl. Ross. Acad.Nauk,
to appear.[16]
S.V.Ludkovsky.
Gaussianquasi-invariant
measures onloop
groups andsemigroups
of real manifolds and theirrepresentations. IHES, Bures, France, preprint IHES/M/97/95.
[17]
S.V.Ludkovsky. Quasi-invariant
measures on non-Archimedean groups andsemigroups
ofloops
andpaths,
theirrepresentations.
IHES/M/98/36.
[18]
S.V.Ludkovsky. Quasi-invariant
measures on groupsof diffeomorphisms
of Schwarz class of smoothness for real manifolds.
IHES/M/97/96.
[19] S.V.Ludkovsky. Quasi-invariant
andpseudo-differentiable
measures on anon-Archimedean Banach space. Intern. Centre for Theoret.
Phys.
Tri-este,
Italy. Preprint (http://www.ictp.trieste.it) IC/96/210,
October1996.
[20]
S.V.Ludkovsky. Quasi-invariant
measures on a non-Archimedean group ofdiffeomorphisms
and on a Banach manifold. ICTP.IC/96/215,
Oc-tober, 1996.
[21]
S.V.Ludkovsky. Quasi-invariant
measures on groupsof diffeomorphisms
of real Banach manifolds. ICTP.
IC/96/218, October, 1996.
[22]
M.A. Naimark. Normedrings (Moscow: Nauka, 1968).
[23]
Yu.A. Neretin.Representations
of the Virasoroalgebra
and affinealgebras.
in:Itogi
Nauki i Tech. Ser. Sovr. Probl. Math. Fund.Napravl(Moscow: Nauka) 22(1988),
163-230.[24]
E.T.Shavgulidze.
About one measurequasi-invariant
relative to an ac-tion of a
diffeomorphisms
group of a finite-dimensional manifold. Dokl.Akad. Nauk SSSR.