Rapidly Decreasing Functions in Reduced C*-Algebras of Groups
Author(s): Paul Jolissaint
Source: Transactions of the American Mathematical Society, Vol. 317, No. 1 (Jan., 1990), pp.
167-196
Published by: American Mathematical Society
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TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 317, Number 1, January1990
RAPIDLY DECREASING FUNCTIONS
IN REDUCED C* ALGEBRAS OF GROUPS
PAUL JOLISSAINT
L on F the
Let F be a group.Weassociateto anylength-function
to L ), which
functions
on F (withrespect
space HL (F) ofrapidlydecreasing
toruswhen
coincideswiththespaceof smoothfunctions
on the k-dimensional
F = Zk. We say that F has property
(RD) if thereexistsa length-function
L on F suchthat HO (F) is containedin thereducedC*-algebraC,*(F) of
(RD) withrespectto someconstructions
of property
F. We studythestability
of finiteindex,semidirect
and amalof groupssuchas subgroups,
over-groups
groupshave property
gamatedproducts.Finally,we showthatthe following
(RD):
growth;
groupsof polynomial
(1) Finitelygenerated
of anyhyof thegroupof all isometries
(2) Discretecocompactsubgroups
perbolicspace.
ABSTRACT.
INTRODUCTION
on the k-dimensional
ConsiderthealgebraC( Tk ) of continuousfunctions
torus,whichis also the C*-algebraof the group Zk. It containsthe dense
subalgebraC??(Tk) of smoothfunctionson Tk. Derivationis possibleand
usefulon C' (Tk), but is not allowedon the wholeof C(Tk). Our aim is
to developa notionof smoothfunctions
in thereduced C*-algebrasof other
of C?(T"k) by Fourier
groups. The idea is to considerthe characterization
whichis a function
series:if f c C( Tk) andif f denotesitsFouriertransform,
o
k
k
on Z , recallthatf belongsto CO (T ) if and onlyif f is of rapiddecay.
on F, we denote
Now, if F is any groupand if L is a length-function
on F withrespectto L.
by HL (F) thespace of rapidlydecreasingfunctions
ExamplesbelowshowthatHt (F) is notalwayscontainedin thereducedC*(RD) ifthereexistsa lengthalgebraC,*(F) of F. We saythat F has property
functionL on F suchthat HL (F) is containedin C,*(F). Some aspectsof
property
(RD) havebeen alreadystudiedin [Haa, FP, and Pi].
definitions
Thispaperis organizedas follows:Chapter1 containselementary
and property
and resultsconcerning
(RD). Proposition1.2.6
length-functions
resultsince it givesa technicalconditionequivalentto
is the firstimportant
ReceivedbytheeditorsOctober1, 1987and,in revisedform,May 31, 1988.
46L99; Secondary43A15.
1980Mathematics
(1985 Revision).Primary
SubjectClassification
(D 1990 American Mathematical Society
0002-9947/90 $1.00 + $.25 per page
167
PAUL JOLISSAINT
168
usedlater:itis an adaptationofLemmas
property
(RD) whichwillbe frequently
1.3 and 1.4 of [Haa].
(RD) withrespectto some
of property
We studyin Chapter2 the stability
of finiteindex,
over-groups
of groupssuchas subgroups,
classicalconstructions
the following
semidirectproductsand amalgamatedproducts. In particular,
theoremgeneralizesthecase of nonabelianfreegroupsof finiterankdue to U.
Haagerup[Haa]:
(RD) thenso does theirfreeproduct
TheoremA. If F1 and F2 haveproperty
r1 *F2.
(RD). First,
Chapter3 is devotedto examplesof groupspossessingproperty
generatedamenablegroups:
we deal withfinitely
groupand let L denotetheword
generated
B. Let F be a finitely
Proposition
on F.
length-function
growth
thenHt (F) is containedin 11(F), and thus
(1) If F is ofpolynomial
in C*(F) .
a fortiori
(RD) ifand onlyifitis ofpolynomial
(2) If F is amenablethenF hasproperty
growth.
(RD) when n > 3.
This resultimpliesthat SL(n, Z) does nothaveproperty
(RD) fordiscontinuIn thesecondsectionof Chapter3, we studyproperty
and exceptional).
spaces (real,complex,quaternionic
ous groupsof hyperbolic
definedas follows:let xo be a pointof
There,we choose a length-function
thehyperbolic
space,and definethelengthL(g) of theisometryg to be the
distancebetweenxo and g(xo). We show
subgroup
ofthegroupofall isometries
TheoremC. If F is a cocompactdiscrete
ofa hyperbolic
space,thenHL(F) is containedin C>F).
(RD). Though HLt(SL(2,Z)) is not conThus such a grouphas property
showsthatSL(2, Z) has property
argument
tainedin C> (SL(2, Z)), a different
(RD) (Corollary2.1.6).
(RD)
studyproperty
We endthispaperwithan Appendixin whichwe briefly
fornot necessarily
discretegroups.We provethatsuch a grouphas property
(RD).
(RD) if it containsa discretecocompactsubgroupwithproperty
of
thefirst
partof mydoctoralthesisat theUniversity
This workconstitutes
of Pierrede la Harpe. I wouldlike
Geneva,carriedout underthesupervision
to thankhimwarmlyas wellas GeorgesSkandalis,UffeHaagerupand Vaughan
F. R. Jonesforfruitful
discussionsand suggestions.
1. RAPIDLY
DECREASING
FUNCTIONS AND PROPERTY
(RD)
concerning
Let F be a group.Wereferto Chapter7 of[Ped]forthenotations
standardspacesand operatorsalgebrasassociatedto F.
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
functions.
and rapidlydecreasing
1.1. Length-functions
on a groupF is a map L: F
1.1.1. Definition.A length-function
fying:
(i) L(gh) < L(g) + L(h), forall g,h E F;
(ii) L(g) =L(g 1),for everyg E F;
of F.
(iii) L(1) = 0, where1 denotestheidentity
-*
169
R+ satis-
on F, we saythat L2 dominatesL1 if
If L1 and L2 are length-functions
thereexist a,b c R+ suchthat L1 < aL2+ b. If LI dominatesL2 and L2
dominatesL1 , then L1 and L2 are said to be equivalent.
on F and r E R+; thecrownof radius r is the
Let L be a length-function
set
Cr,L = {g E r; r - 1 < L(g) < r},
functionof Cr ,L
and Xr,L denotesthecharacteristic
The ball of radius r is theset
BrL = {g=F;L(g)
< r}.
Notethat BO,L is a subgroupof F whichmayhavemorethanone element.
and let S be a finiteset
generated
1.1.2. Example. Supposethat F is finitely
of g E F withrespectto S is theleast
of F. The algebraiclength
ofgenerators
as a productof n elements
integern suchthat g can be written
nonnegative
of SuS I . We denotethisnumberby IgIs and geta mapfromF to N which
set of F, it is readily
If S' is anotherfinitegenerating
is a length-function.
associatedto S and S' are equivalent.Most
thatthelength-functions
verified
and will speak
of the timewe willnot specifyanyfinitesystemof generators
on F. If Io is a subgroupof a finitely
about thealgebraiclength-function
on
to 0O of the algebraiclength-function
generatedgroup F, the restriction
not
F is a length-function
finitely
generatedin
on FO. It is knownthat 0O is
general.A simpleexampleis givenby thesubgroupF of GL(2, R) generated
by
s
=)(
and
52=(0
I)
and its subgroupFO generatedby {s S2s7k k c Z} whichis isomorphicto
Z[l1/2].
1.1.3. Example.Let X be a metricspace withbase point xo c X and let F
on X. Defineforeveryg c F
be a groupof isometries
(g) = d(xo, g(xo)).
LXO
If xl
on F sinceeach g c F is an isometry.
Then L. is a length-function
thefollowing
inequalitiesforevery
is anotherpointof X, L,o and Lr satisfy
g c F:
(g) + 2d(xo,xl).
(g)-2d(xo, xl) < Lxl(g) < LXO
LXO
leftto thereader.
lemmais easyand consequently
The proofofthefollowing
170
PAUL JOLISSAINT
1.1.4. Lemma.Let L be a length-function
on a groupF.
(1) Wehave JL(g)- L(h)I < L(gh) foreveryg, h c F.
(2) If F is finitely
generatedthen L is dominatedby thealgebraiclengthfunction.
(3) (Peetre'sinequality).If s is a real number,
one hasforall g, h c F
(1 + L(gh))s < (1 + L(g))Isl (1 + L(h))s.
(4) If L' is a length-function
dominatedby L, thereexistsc c R+ suchthat
<
>
r
for
every
whereIBr,LI denotesthecardinalof Br L .
1,
I
I
Br,L
IBcr,L'
1.1.5. Remark.The converseof assertion(4) in Lemma 1.1.4 is falsein general. In fact,if F = Z e z and if L1 and L2 are definedby
LI (x,y) = |x| + log(1+ IyI)
and
L2(x,y)
= LI(y,x),
then JBrL I = JBr,L2Iis finiteforeveryr > 0, but Li does notdominateL2
whichdoes notdominateLi .
Let us nowgivethemaindefinition
of thiswork:
1.1.6. Definition.Let L be a length-function
on F7.
(1) If s c R, the Sobolevspace of orders (withrespectto L) is the set
HLs(1) of functions4 on F suchthat4(1 + L)s belongsto 1 (F) .
(2) The spaceof rapidlydecreasing
on F (withrespectto L) is the
functions
c HL(F) , set
set HL??(r) = nSERHL(F) * If s c R and
(41C)2,s,L =
, (g)Z(g)(1
+L(g))
gEr
and 1112sL
V'2)2,s,L
Withthe above innerproduct,HL(1) is a Hilbertspace, and HL??(r) is a
Frechetspace forthe projectivelimittopologyinducedby the inclusionsof
we denoteby
generated,
HLt (F) in HL(F), foreach s c R. When F is finitely
Hs () and H??(F) theabovespacesassociatedto thealgebraiclength-function
on F. We denoteby (.)2 25 and 1 * 112
innerproductand
2, thecorresponding
norm.
1.1.7. Remark.Let Li and L2 be twolength-functions
on F. If L2 dominates L1 then Hj (F) is containedin HL (F) foreverys > 0. In particular,
if F is finitely
thenH' (F) is containedin HL?(F) foreverylengthgenerated
functionL on F. We finally
on F and
remarkthatif L is a length-function
E
if s R, then H[s (F) is thedual of Hs(F) forthebilinearform
(a'g)
gEr
)(g)
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
171
1.2. Property
(RD). Let F be a group.The groupalgebraof F is denotedby
CF and is the set of functions
withfinitesupporton F. We recallthat CF
on 12(F) and thatthereducedC*-algebraof
actsfaithfully
left
convolution
by
F, denotedby C,*(F), is thenormclosureof CF in B(l2(F)) . Furthermore,
everyelementa of C, (F) can be identified
withtheimageof 6 by a, where
J is the characteristic
functionof { 1} in F. We shallthusconsiderC,*(F)
as a subspaceof 12(F). If (o E C*J(F),we denoteby 11oHl
the normof fo in
Cr
(r)~~~~~~~~~~~~~~~
C*(IF).
1.2.1. Definition.A group F is said to have property
(RD) if thereexistsa
F
length-function
L on suchthat HL?(F) is containedin Cr*(F)
(1.2.2.) Remark.(1) By the closed graphtheorem,F has property(RD) if
L on F and twopositivenumbersc
and onlyif thereexistsa length-function
and s suchthat klol< C11Iko12jsL foreveryo E CF.
(2) If F is finitely
generatedand possessesproperty
(RD) then H??(F) is
containedin C, (F) by Remark1.1.7.
1.2.3. Example. (1) If F is the infinite
cyclicgroupthenit is easilyverified
that F has property
(RD); we haveforeveryp E CF
l(Pl<
110112,1
(2) Let -FN be the (nonabelian)freegroupof rank N > 2. Then FN has
foreach
property
(RD) sinceLemma1.5 of [Haa] givesthefollowing
inequality
function(p withfinitesupporton FN:
? 2119112,2
koH11
More generally,
we will showin Chapter2 thata freeproductof twogroups
withproperty
(RD) has itselfproperty
(RD).
on F suchthatHL?'(F) is contained
1.2.4. Lemma.If L is a length-function
in Cr*
(F), thenthereexist c and s in R+ suchthatone hasforeveryt E R
and (0, VJ HL (1)
11(P* V112,t,L - 11(0112,s+th
,Li l/112I,t,Lv
In particular,HL?(F) is a convolution
algebra.
Proof.By Remark1.2.2,thereexist c, s E R+ suchthat
11(P* v'112< CI01I2,s,JLil112'
for (0,
v E
HL??(F). Let us define
(t(h)
=
19(h)l(I + L(h))
and
ylt(h)=
qi(h) (l +L(h))t
PAUL JOLISSAINT
172
for h c F. Then (Ot and
belongto Ht (F) and theysatisfy
it
IPhtHI2,s,L =
and
II2,s+jtj,L
Iv'J12=
112,t,L
Then one gets,usingPeetre'sinequalityof Lemma 1.1.4
Io
*
V/112,t,L ? II(*t
?
'i'1H2
CIIII2,s+jtj,LH
Q.EID.
L
2,tI
(RD)
forthestudyofproperty
motivations
In fact,one ofthemostimportant
is givenby the followingresultof K-theorywhichshowsthatthe subalgebra
HL (F) is in somesense"large"in C,*(F). The resultis due to A. Connes,and
we givea proofof it in [J3].
suchthat
(RD) and if L is a length-function
1.2.5. Theorem.If F hasproperty
(F)
induces
in
of
(F)
then
the
inclusion
in
C*
(F),
is
contained
Cr*
HL
HL?(F)
K1
=
1.
for
i
K!
0,
an isomorphism
from (HL??(r)) onto (Cr>()),
(RD).
whichare equivalentto property
Now we establishsome properties
Let A(F) denotetheFourieralgebraof F (cf. [Ey]) and let B (1) be theset
of F whichare weaklycontainedin
of coefficients
of unitaryrepresentations
(F) forthebilinearform
theleftregularone. B, (F) is thedual space of Cr*
(i"
v')=
Z 6(g)Y/(g)
gEr
of the
by thecoefficients
for (0-c CF, V c B2(F). Since A(F) is constituted
A(F) is a closedsubspaceof B,(F) equippedwith
leftregularrepresentation,
thedual norm.
on a groupF. Thefollowing
1.2.6. Proposition.Let L be a length-function
are equivalent:
properties
(1) HL(F) is containedin Cr,(F);
embeddedin HLS();
(2) thereexistss > 0 suchthatBs(F) is continuously
embeddedin HLs(r);
(3) thereexistss > 0 suchthatA(F) is continuously
(4) thereexistsc > 0 and r > 0 suchthatif k,l, m belongto N, if (p
in Ck,L and CIL respectively,
and VI belongto CF and are supported
one has
? cllRoHl2,r,LHVH2
ifIk - 1 < m < k + I
*( VI)XmLH2
and IVo * V)Xm,L112 = 0 fortheothervaluesof m.
Proof. (1) => (2) By Remark1.2.2,thereexistss > 0 suchthat HL(F) is con(2) is obtainedby duality.
embeddedin Cr*
(F). Thus,property
tinuously
(2) => (3) is immediate.
(3)= (1) Letus fix(
Let V/be definedby
and c, i cl (F) suchthat 1H2?1
c,
CCF
V(g) =(E
*)(gnA),
foreveryg cF .
HH2<1.
RAPIDLY DECREASING
IN REDUCED
FUNCTIONS
C*-ALGEBRAS OF GROUPS
173
Then / cA(E7) and IIY'IA(r) ? II4II2II5II2
< 1 (cf. [Ey]). It is readilyverified
= ((0, V/),which gives
that (( *
(P * 41011 _ 11(P112,s,L11V1112,-s ,L
?
CII(112,s,LI11VI1A(r)
< C 11012s,L'
(1) => (4) is immediate.
(4) (1) Let (0, Vgc CF; supposefirstthat (0 is supportedon Ck ,L for
some k c N. Put V1= V *%IL foreveryI c N. One has by hypothesis
11(( * ?/1)Xm,L1I2 < C(l
and
((0 * v/1)X%n,L=
Thenone gets
II(
V)Xm,L
+
if Ik-11 < m < k + I
k)rII11211V'1112
0 otherwise.
Z
112 ?
II(
* V/I)Xm,L
112
1>0
m+k
< c(l + k)r119112
I
I=lm-kl
IIV1112
2 min(k ,m)
< c(l + k)r11(0112 E
IIV/m+k-/112
1=0
(2min(k,m)
Z
< c(l + k)r (2k + 1)112112
1/2
J
IIY'm?k-I112
1=0
Consequently,
I
*
Z
l'IL
11 =
?
m>0
I('
*
12
cl2C
+ k)
(Il+)2r+l,,
2 min(k ,m)
II2I
P112
I/+-12
m>0
2
C2(1
+ k)2r+2
Finally,if the supportof
1=0
III2I I2
is arbitrary,
then o =
q
Zk>0
( *Xk,L* One has
1101?
<
< C2 (1
k>0
EII|kl
k>0
<3
? c3
(P
11k
l2 (
k>0
< C4lk0ll2, r+2L
One can take s = r+2.
Q.E.D.
+
+
+k)
k)
)2(r?2))
119k112
)
/
9k'
where Pk
=
174
PAUL JOLISSAINT
1.2.7. Remark.Let L be a length-function
on F. If L is boundedthen
HL (F) = 2(F). And by [Raj], 12(F) is an algebraif and onlyif F is finite.
More generally,HL (F) is not an algebrain general,even in the case of the
as thefollowing
algebraiclength-function,
exampleshows:
1.2.8. Example.Let C denotethe infinitecyclicgroupwrittenmultiplicativelywithidentity1 and generatora, and let A = Z[C] with cj the characteristicfunctionof aj. Then C acts on A by xf(y) = f(x1ly), so that
anJ , j) = gj+n -
Let F = A x C be theassociatedsemidirect
product.F is generated
by the
finiteset S = {(.co 1),,(0,a)}, thoughA is not finitely
generated(compare
withExample1.1.2). Fromnowon we willidentify
theelementa of A with
F
theelement(a, 1) of and we willdenoteby lal+ thelengthof (a, 1) with
respectto S. If p is a positiveinteger,
set
p~~~~~
SP lacA;a=
i aE
=EaijE
{0,
,ap =1I
Let a be a positiverealnumberand letus definethefunction(0 on F by
(q(g
0,
if g0up>, SP,
ifg EU S
{ap/2
1.2.9. Proposition.
If l/V'5< a < 1/2 then(0 E Ho'(F) but (0 * ( does not
belongto anyspace Hs(F) ,s E R.
Proof.If a = Ep? aJe belongsto Sp, then Iar < 3p + I since
(a, 1) = (,co,1)a, (0, )(t0,
a)
l)a,(0
l)aP(0, C)-P
..
If s > 0, then
2i'i
? E20)P
(2a{p(3pP+ 1) 2s <00o
f11,s<
p>l
since 0 < a < 1/2 and ISPI = 2P. This showsthat ( belongsto H??(F).
Let us nowfixa positiveintegerp. If a C A is of theform
a
=
p-l
E a,
J=o
+
2
to {0,
witha1 C {0, }, let l(a) denotethenumberof j belonging
that
a1
One
has
for
an
element
a
such
= 1.
such
zp * ((a)
=
E
b,cESp
a=b+c
9(b)
f(c) .
...
,p - 1}
FUNCTIONS IN REDUCED
RAPIDLY DECREASING
C*-ALGEBRAS OF GROUPS
175
If l(a) = k, thereare exactly2k distinctpairs (b , c) in Sp x Sp satisfying
a = b + c. It followsthat ( * (o(a) = 2ka1p forsuchan elementa. Put
p-i
= k}
A; a= EaJ6J +2cp, aJ {O, 1},(a)
Sp k{a
j=0
= {g c
The cardinalof Sp k is equal to (P), and S k is containedin B
<
3p+2}.
F;lgl
One gets ( + lalr) 2s> 3 2(1 +p 2s ConseLet s > 0 and a c SPk
quently,
11 *qll2
>
(o * o(a))2(1
E
E
+
p? 1 k=O aESp,k
> 3 2s J(
_2
> 32s
+p)- 2sE
2p
k=O
pP>
EZ
(P)4k
lalr)2s
(aV35-2p_
P)2
as soon as a > 1/v"5. Q.E.D.
2. PROPERTY
(RD)
AND SOME CONSTRUCTIONS
OF GROUPS
easyfact:
remarkthefollowing
(RD) and extensions.Let us first
2.1. Property
of F. If F has
be a subgroup
2.1.1. Proposition.Let F be a groupand let
on F
if L is a length-function
(RD) thenso does 0. Moreprecisely,
property
suchthatHL' () is containedin C,*(1), thenHr(Fo) is containedin Cr (O),
of L to FO.
whereLo is therestriction
rO
Now considertwo groups G and F, and let E be an extensionof G by F.
Let 1 -* G -* E "F -r 1 be the correspondingexact sequence. Choose a set-
a: F
theoreticcross-section
a function f: F x F
-+
E of 7r suchthat v(1) = 1. This determines
G measuringthe failureof a to be a homomorphism,
namelyf(YI Y2) = a(y1)a(y2)a(y1y2) forall YI1Y2 c F. In addition,let
p(y) be theconjugationby v(y) in G: p(y)(g) = a(y)ga(y) ' . For a c F,
-)
of F associatedto a. Then the
let Ad(a) denotethe innerautomorphism
functionsf and p are relatedby
(R 1)
p(/3)p(y)
=
Ad(f(3, y)),p(y)
and
(R2)
f(y1,y2)f(y1
Y2Y3).
y21Y3)= P(Y1V)(f(y212y3))ff(Y1
(See [Bro,p. 104].)
PAUL JOLISSAINT
176
We shallidentifyE with G x F equippedwiththeproduct
(g1 I1 )(g2
Y2) = (g1 P(Y1)(g92)f(Y1
Y2) 1 YI2)*
2.1.2. Lemma. Let E, G and F be as above. Suppose that thereexist lengthsuch that
functionsLO,L1 and L on G, F and E respectively
(i) HLo(G) is contained in C,*(G) and Ht (F) is containedin
(ii) thereexist c and r in R1 such that
LO(g) + LI (y) < c *L(g,
y)r for every(g,
T)
y) E EE.
Then HL? (E) is containedin Cr (E).
Proof. Choose firsttwo positive constants d and s such that
< dllHpoll2
IIHpolI
,S,Lo for every(poE CG
and
lol II < dIIolH,
11S2,,L, for everyo1, EcF
If
(0
.
and V belongto CE, one gets
2
Z
-
2
Z
(If1 I)(h 1g)f(/f ,)/31
((h,/3)yI(f(/,/3,)
(g,y)EE (h,/3)EE
Iy)
2
=
,E
yErgEG
,
,
EF hEG
fofl(h)vfisy(h
g)
2
2
YEF PEF
where
(f (g)
=
O(g,3)
and
y'fi (g)
=
(f (/l
/3)1-p(/3p1)(g)f(/3l y) /'-'y)
It followsfromthetriangleinequalitythat
| |Sq/
|| y-E
<d2
(
qp{#1
V11
2I<
= d2 11o*
where (0(l)
=
12,s,Lo and v"()
]
? d2
ISV/f,lv2sL
=
d
Gl</(g
(Zh
2
2
112
2)12we
haveus
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
177
Finally,condition(ii) impliesthat
< CHII(OII2,2rs,LIl/I2.
* yV112
Q.E.D.
IIDP
We willfirsttreatthecase when G or F is finite.Let us givethefollowing
lemmawhichwillalso be used in thefollowing
paragraph:
2.1.3. Lemma.Let E be a group,let G be afinitesubgroupof E and L a
on E. Thenthereexistsa length-function
length-function
L' on E withthe
following
properties:
(1) L' is equivalentto L and takesintegervalues;
(2) L'(gxh) = L'(x) forall x c E and g, h E G;
(3) G = {x E E;L'(x) = O}.
Proof.Let us definesuccessively
fourlength-functions
L1, L2, L3 and L4 on
E :
LI (x)
= Tif
[L(x)] + I
then L1 satisfies
condition(1).
Now,thefunctionL2 definedby
L2(x)
=
>
L(x) =0,
if L(x) 0& ;
L1(gxg1)
gEG
is a length-function
on E satisfying
condition(1) and is suchthat L2(gx) =
L2(xg) forall xEE and geG.
Set L3(x) = ming,hEG L2(gxh); thenL3 is a length-function
on E satisfying
conditions(1) and (2).
Finally,if
L(x
{
x
E
G,
then L4 is a length-function
on E, and L' = L3 + L4 satisfies
conditions(1),
and
(2)
(3), Q.E.D.
2.1.4. Proposition.Let E be an extensionof G by F as aboveand suppose
thatG isfinite.Then E has property
(RD) ifand onlyif F does.
Proof. Supposefirst
thatF has property
(RD) and let L1 be a length-function
on F such that HL(r) is containedin C>(F). For each (g,y) c E set
L(g, y) = LI (y) E
Then L is a length-function
on E satisfying
theconditionsof Lemma2.1.2
whichimpliesthatE has property
(RD). Conversely,
supposethatE has propon E suchthat
erty(RD) and let L be a length-function
(a) HL(E) is containedin Cr,(E);
(b) LIG= O (see Lemma2.1.3).
Set LI(y) = maxgEGL(g,y).
Thanks to condition (b), L1 is a length-
on F. If (0 and V belongto CF, definefunctions(o' and y' on E
function
by
(0(g, y) = ( (y) and V'(g, y) = yV(y).
178
PAUL JOLISSAINT
One gets
1~~~~
131f*
V11,2=
2
C I
12
f I2,
I
IGI1
2
<C 1
1112
W12
112
V11
? IGIIIOI2,s,LIIY/I2
forsuitablepositiveconstantsc and s.
Q.E.D.
2.1.5. Proposition.Let E be a groupand let Eo be a subgroup
index
offinite
of E. If Eo has property
(RD) thenso does E.
Proof.Define G = nxeE xEox 1; it is a normalsubgroupof finiteindexof
E and it is containedin EoI It followsfromProposition2.1.1 that G has
property
(RD). Let then Lo be a length-function
on G suchthat H2 (G) is
containedin C> (G) . Set 17= E/G and define
k(g,y) = maxL0(p(,B)(g)f(fl,y)) forevery(g,y) E E.
/JEF
of the chapter,it is easily
Using relations(R 1) and (R2) of the beginning
shownthat
k((g1,
Set
L yl)(g2j
k+
Y2))
< k(g, ,yl) + k(g2,
and kiG > Lo.
Then L is a lengthk, wherek(x) = k(x'1) .Y2),
=
function
on E satisfying
theconditionsof Lemma2.1.1. Q.E.D.
2.1.6. Corollary.If1 is a finitely
generated
discrete
subgroup
of SL(2, R) such
thatSL(2, R)/F is notcompact,thenF hasproperty
(RD).
Proof.By Lemma8, p. 154 of [Sel], F containsa torsion-free
subgroupFO of
finite
index.ThenthesurfaceSO(2)\SL(2, R)/F0 is noncompact,
namelyopen,
and its fundamental
groupJ7Ois a freegroup. By Example1.2.3(2), FO has
property
(RD) and by Proposition2.1.5, F has property
(RD), too. Q.E.D.
2.1.7. Remark.Let E be a groupand let G be a subgroupof finiteindexof
E providedwitha length-function
no length-function
LoI Thereis generally
L on E whoserestriction
to G coincideswithLoI In fact,take G = F2, the
freegroupon thetwogeneratorsx and y, and let oa be theautomorphism
of
G definedby
Then a
2
a(x)
1. Let F
=
{l,}
= xy,
a(y)
and let
= y
E = G x F be the corresponding
semidirectproduct. Suppose thatthereis a length-function
L on E whose
restriction
to G coincideswiththenaturalalgebraiclength-function
on G. We
wouldgetforeach m E N
= L(a(x'xn), 1) = L((l ,t)(x'n , l)(I ,a))
2m = Ias(x,n)I
< 2L(l, a) + Ix'nl= 2L(l, a) + m,
whichgivesa contradiction.
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
179
Finally we consider extensionsof finitelygeneratedgroupsby finitelygenerated ones.
Suppose that G and F are finitelygeneratedand let S = S 1 and l = 1
be finitesystemsof generatorsof G and F respectively.It is easily shown that
} generates E.
the finiteset T = {(s, 1); s E S} U {(, z);zE
2.1.8. Definition. (1) Let a E Aut(G). The amplitude of a (with respectto
S) is the number
a(ce) = max a(s)I.
(2) A map 0: F -* Aut(G) has polynomialamplitudeif thereexistpositive
constants c and r such that
a(0(y)) < c(1 +
1yI)r
for everyy E F.
(3) A map f: F x F -* G has polynomialgrowthif thereexistpositive
constants c' and r' such that
If(Y1 Y2)I < C'(1 ? I)r(1
+ IY2)
forall y1,y2EF.
and thatthey
generated
2.1.9. Proposition.SupposethatG and F arefinitely
(RD). Let E be an extensionof G by F and supposethatthere
haveproperty
is a pair ofassociatedfunctions(p, f) suchthat p has polynomialamplitude
(RD).
growth.Then E has property
and f hagpolynomial
on G, E and F
Proof. It sufficesto show that the algebraic length-functions
satisfythe conditions of Lemma 2.1.1. There exist positive constants c and r
such that
a(p(y)) < c(1 +?YI)
and
If(Y1'Y2)I < C
C(1 ? I1
1 + IY21)
forally,y1,y2zF.
Let then (g,y) E E and supposethat I(g,y)l = n > 0. Then thereexist
-I
(glY ), ***1(9n yn)ETUTT suchthat
n
(g,Y) = J7(gj,yJ)=
fn
r1P(/3J-1)(9?)
J=1
j=1
wheref3o= l and
fk
= Y1
Thus
Yk
(a(p(3j-
1
1)) gjI + if(3J-
J=1
< clnr+?= c11(g,
and IYI< n = I(g,y)I Q.E.D.
J
for k > 0.
n
Ig <
y
fln
)f-1'YJ)
y)r+l
YJ))
PAUL JOLISSAINT
180
2.1.10. Corollary.Supposethata is an actionof F on G ofpolynomialamproduct
(RD) thenso does thesemidirect
plitude.If F and G haveproperty
Gx(IF.
2.1.11. Example.Let F, and F2 be thefreegroupsof rank1 and 2 respecof F2, let F1 act on F2 in the
tively.If x and y are thenaturalgenerators
following
way:
and a(y) =x.
a(x) = xyx'
F2 xa F1
amplitudeand consequently
thata has polynomial
It is easilyverified
(RD). But thisgroupis a normalsubgroupof finiteindexof the
has property
(RD),
braidgroupB3 (see [BZ, Chapter10]). ThereforeB3 possessesproperty
too.
products.Let F1,F2 and A be groups
(RD) and amalgamated
2.2. Property
suchthat
(1) FJ admitsthepresentation(EJIRJ),for j = 1,2;
(2) thereexistsan injectivehomomorphism
fj fromA to rj, for j=
1,2.
We recallthattheamalgamatedproductof 1l and F2 over A is thegroup
thefollowing
presentation:
admitting
(X1UX21R1U R2,f1(a) = f2(a) foreverya E A).
It is denoted by F1
*A rF2.
(See [LS].)
ofleftA-cosetsin F .. Theneveryelement
Choosea set Sj ofrepresentatives
written
as a reducedword g = si ... ska, where
F2
can be uniquely
g of FI *A
aEA and i1: ij+I foreveryj= 1,... ,k-.
sjeSI\{1},
integers;
put
Let k and 1 be nonnegative
Ak = {g EF] *A F2; g= sI .ska as a reducedword}
and
Ek l(g) = {(hl,h2) EAk x A,; h,h2= g}
where g is an elementof
rF
*A F2.
lemmais takenfrom[Pi, Lemma3.1]:
The following
integerssuch that m =
2.2.1. Lemma.Let k , 1, m and q be nonnegative
k + 1- q, and let g an elementof Am. If g = s, ... sa is its reducedform,
one has
Then
(1) If q = 2p is even,set g1 = SI Sk_p and g2 = Sk-p+l .ma
=
such
that
x
w
E
=
exists
there
E
Ak
(h1
,h2)
(g)
Ap
hi giw and
A,;
EkJ
h2=W
g2}(2) If q = 2p?+ is odd,set g, = S1 ... Sk-p-I and g2 = Sk-p+l .*ssna. Then
Ek 1(g) = {(h, ,h2) E Ak x A,; thereexist w E Ap and vl, v2 e A1 suchthat
hi=ggviw, h2=w
v292
and v
V2= Sk-p
theThe restof thisparagraphwillbe devotedto theproofof thefollowing
orem:
RAPIDLY DECREASING
FUNCTIONS
IN REDUCED
C*-ALGEBRAS OF GROUPS
181
2.2.2. Theorem.TheamalgamatedproductF1 *A F2 hasproperty
(RD) in the
following
cases:
(1) r- and F2 haveproperty
(RD) and A isfinite;
(2) A has property
(RD) and is centraland offinite
indexin rF and F2.
2.2.3. Corollary.(1) If F1 and F2 possessproperty
(RD), thenthefree
product
F1 * F2 has property(RD) too.
(2) Let p and q be positiveintegers
suchthat (p, q) = 1 . Thenthegroup
F = (x, ylxp= yq) has property
(RD).
In both cases of Theorem2.2.2, we have firstto definea suitablelengthfunctionL on 1-*A f2
(1) If r1 and F2 have property
(RD) and if A is finite,choose a lengthfunctionLi on Fj suchthat
(i) Li takesinteger
values;
(ii) Li(agjb)=Li(gi)
forall gjeFj
(iii) {gj E j;Lj(gj) = O}=A;
(iv)
HL (IJ)
and a,beA;
is contained in Cr (rj
The existenceof sucha length-function
is provedin Lemma2.1.3.
If g E 1 *A F2, let g = s1 *stna be thecorresponding
reducedword,with
Set
\{
1}.
eS1
si
L(g) = L,,(sl) + + Lim (stn)
It is easyto verifythat L is a length-function
on 1- *A F2.
(2) If A has property
(RD) and is centraland of finiteindexin 1- and F2,
choosefirsta length-function
properties:
Lo on A havingthefollowing
(i) Lo takesinteger
values;
(ii) HLO(A) is containedin Cr (A);
(iii) foreveryelementa E u_ {(X E A; thereexist s, t, u E Si withst =
ua}, one has LO(a) < 1.
The existenceof Lo is ensuredbythesame arguments
as in Lemma2.1.3.
Set then for each g E F1
*A
F2
K(g) = m + Lo(a)
whereg = s. s,na is thereducedwordassociatedto g . UsingLemma2.2.1,
it is easilyshownthat K(gh) < K(g) + K(h) forall g, h E 1- *A F2. Finally,
defineL = K + k. Then L is a length-function
on 1- *A r2 suchthatits
restrictionLJ to Fr satisfiesHL (FC)c CC(>i).
Here is thecrucialstepin theproofof Theorem2.2.2; it is an adaptationof
Lemma 1.3 of [Haa]:
2.2.4. Lemma.Supposethat 1,F2 and A satisfyone of the conditionsof
Theorem2.2.2. Thenthereexist c and r > 0 suchthat:if k , 1,m E N satisfy
182
PAUL JOLISSAINT
ik - 11< m < k + 1, if (p and V E CFI
respectively,
one has
* A r2A2are supported
in
Ak and
* V)XAm112< CIkPII2,r,LIIVfII2'
||(
whereL is thelength-function
above.
defined
Proof.The proofis decomposedintotwopartscorresponding
to theconditions
of Theorem2.2.2.
Firstpart. Case 1.1. Supposethat m = k + 1 - 2p. UsingLemma2.2.1 and
similararguments
as in theproofof Lemma 1.3 of [Haa], one verifies
that
< Ni1
II(P* V) XA,n112
V11/2
11Q211
where N =IAI.
Case 1.2. Suppose now that m= k + I - 1. If g = s. sma belongs to Am,
set g1= S
sma. By Lemma2.2.1,
Sk-I and g2 =Sk+
(p )SE
E
bEA (VI,V2EEII1(Sk))
since E1,
Sk
F
F1
v E F.
.
((gllvb)
,v2)EAl
(sk)={(vl
and set
(Pgl b(V)
Then one gets
xAl; Vv2=
= p(g vb) and
9 * tu(g)=
i(b Iv2g2)
Let kE{l,2}
Sk}l
be suchthat
(v) = Vg(b'vg2) forevery
lb-,
(kr,k (("g ,b) 1lb-g2)(sk)
bEA
Finally
II(?o*
0)xA112< c (max
2
- Ix
(gi,b)EAk_
(~~~~~I
A
||-Vb
1
,g2112)
(g2 ,b)EAI( IxA
cN
1N12IrLIIvfII2,
wherer > 0 is largeenoughto ensurethatHL (Fj) is containedin C*(Fj) for
2.
j=,
Case 1.3. Supposefinally
that m = k + I - 2p - 1 withp > 1. Define
p1(u)
{
(u)EA
/(uw)l
,
if u E Ak-p
)
ifuA Ak-p
0
EIIE
IV|(W-IV)12
I/2
if vEAl-
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
183
Then I9 * V'(g)I < I * VIg(g)foreach g EAm and I'IvI2,r,L < NItlII2,r,L
< Nil112 * Onegets
and 11
Il 112
< 11(91
* VI)XAm
II< C2N21I9112,r,LII
II(( * Y)XAmIL2
V1112
2
usingtheresultestablishedin Case 1.2.
Secondpart. SupposenowthatA has property
(RD) and is centraland offinite
indexin 1l and F2.
Let A denotethe set of elementsg = s.
i. t4ij+ for j < p - l . If f is a functionon
set fg(a) = f (ga) foreverya E A.
sp,where s. E Si.\{l} and
1 *A
F2
and if g e F1
*A 1F2,
Case 2.1. Supposethat m = k + I - 2p. If g E Am it can be uniquelywritten
as g = g,g2a with(g1,g2,a) E Ak x A x A, and we havebyLemma2.2.1
5
* q(g) = E
(AA
(?g,W)/ul-l2)()
uwEAO
wep
It gives
- C3 (1
11(9 * V/)XAM,112
lIgU)12,r,Lo)
(Pg
(g1,w)EAO_ xAP
jj(~*Y)XA, 11? 3
2 kgw12rL
r,
K
S
11
VW 92112)
(1
(92,W)EAO xA?
C3IkOII2
rLI/II2~
since each elementh e A, can be uniquelywrittenas h = aw h'a with
(w, h', a) E A? x A,_ x A.
Case 2.2. Considerfinally
thecase wherem = k + I - 2p -1 and let g E Am
lk
1 xI Al0
5 g2,a) E A?0
with
g = gglsk_pg2a
where(g,Sk_p
pk-p
with
-p-1I x A. Set
k-P-l x A0
1Sk-p2
Skp E A}. Then Lemma2.2.1 allows
A(sk-p) = {(ul , U2) e AOx A?; (u u2
us to decomposeEk /(g) as follows:
Ekj(g)=
U
{(g1u1wb,w1u2g2b aa);w eAA,b and
(u1,u2)EA(sk_p)
a = (UU2)Y 'Skp
E
A}
184
PAUL JOLISSAINT
One gets
II(o*
Y')XA,~II~
? M2((gj
(?9
V)XAin2 -g
,A
F2EAk
<cM2
(u ,u2)EA
,g2)EA? __ xA?
xA? wEA iA (%luilw )
AO A?
(UX
IU2)EA?X
)w-'lu2g2 2)
9UW)1-U
'\glmax
l {1
max12
< c, M
I
EAO
\i9
whereM = IA?I. Q.E.D.
V'1 'I' XA.
~
(9 9)EO
Proofof Theorem2.2.2.
IWEApj
AP-
It suffices to show that the pair
(1
*A
"2,
L)
satisfies
property
(4) of Proposition1.2.6. Considernonnegative
integersk,l1and m
2~~~~~~~
suchthat Ik- li < m ? k?+1,
and let o,qi be elementsof CF1 *A F2 supported
in Ckl and C1 L respectively.Accordingly
to the definition
of L (in both
cases), one has Ck
It follows that
tG/,
~
A,0
~
L CU0=AJ.
f, =
~
Ek=
pO~
Let us first
fixj e {O, ...
.,
/JadW=
an
where
I0
qi =
qJ=
p
%Aand
k} . Lemma2.2.4 gives
m
II(o* t')Xgn,L1||22=
Z II( J* vV)XA1112
,J?p
m
m 2 min(J,P)
? 2(j + 1)C~ll JlI12r
LZ
p=O
1Y+,12
1=O
? C2Iko112r+l LIIY112
Consequently,
jI((o*
S jj(qOJ* )
k
)
LIt2
2
LIt2
k
? C2IIllIII2ZEIi0OJ112,r+l
L
MI
rr2 ? C II2
,L
1112
- Q.E.D
A
RAPIDLY DECREASING FUNCTIONS IN REDUCED C*-ALGEBRASOF GROUPS
185
3. EXAMPLES
We presenttwo familiesof examples:groupsof polynomialgrowthon one
isometries
on theotherhand.
hand,and discontinuous
groupsof hyperbolic
gen3.1. Groupsofpolynomial
growth.In thewholeparagraph,F is a finitely
eratedgroupequippedwithitsalgebraiclength-function.
if thereexist c and r E R+
3.1.1. Definition.(1) F is ofpolynomialgrowth
suchthat lBk I< c(1 + k)r foreveryk > O.
growth
on F, thenF is said tobe ofexponential
(2) If L is a length-function
withrespectto L if thereexist u > O and v > 1 suchthat IBk > u vk for
every k > O.
if it is of exponential
growthwithrespectto
(3) F is of exponential
growth
itsalgebraiclength-function.
3.1.2. Remark.(1) BymeansofLemma 1.1.4,(2) and (4), F is ofpolynomial
L on F and c, r > 0 suchthat
growthas soonas thereexistsa length-function
IBkL? < c(l +?k)r,
foreveryk > 0.
(2) If F is of exponentialgrowth,thenit is of exponentialgrowthwith
on F.
respectto anylength-function
3.1.3. Example.If F is theabelianfreegroupof rank N > 1 and if itsalgeis definedwithrespectto thecanonicalset of generators
braiclength-function
of F, thenwe have foreach k E N
IBLI =
E 2(I)
(I),
growthsincethere
byProposition3.6 of [Wo]. Thus F is clearlyofpolynomial
exist c1,c2 > 0 such that ckN < B < c2kN for every k > 1 . However,
in case F = Z, let L: Z
defined by L(n) =
R+ be the length-function
growthwithrespectto L.
log(1+ lnl). Then F is of exponential
3.1.4. Example.Let FN be thenonabelianfreegroupofrankN > 2, equipped
Then FN is of exponentialgrowthsincewe
withits naturallength-function.
have foreveryk > 0
ICkI
where
Ck =
= 2N(2N-
1)
{ g E FN;Igl = k}.
sub3.1.5. Remark.(1) If F is almostnilpotent(i.e. ifit containsa nilpotent
a
groupof finiteindex) thenit is of polynomialgrowth[Wo]. Conversely,
generatedgroup
deep theoremof M. Gromov[Gro] assertsthateveryfinitely
of polynomialgrowthis almostnilpotent.
exhibitsfinitely
generatedgroupswhichare
(2) In [Gri], R. Grigorchuk
neitherof polynomialnorof exponential
growth.
186
PAUL JOLISSAINT
We are goingto presentsome characterizations
of groupsof polynomial
growthin termsof the spaces Hs(F) and H??(F). Let us introducefirstthe
following
coefficients:
e(F)
=
inf{r> 0; thereexistsc > 0 withIBkI < ck , k E N}
e'(F)
=
inf{r> 0; thereexistsc > 0 withICkI< ck ,k E N}.
and
Notethat F is of polynomialgrowthif and onlyif e(F) is finite.
The proofof thefollowing
lemmais easyand leftto thereader:
3.1.6. Lemma. The coefficiente(F) does not depend on the algebraic lengthfunction.Moreover,e(F) and e'(F) possess thefollowingproperties:
(i) e(F) = lim0sup
and
n log; and
(ii) e'(F) < e(F) < e'(F) + 1.
e'(r)
'F
= lim
u logn
nlm sup
Let us nowrecallbriefly
thenotionof a nuclearspace (cf. [Sch or Tr]).
Let E and F be Banach spaces. A linear mapping u from E to F is
nuclear if thereexist bounded sequences (x' )n>l C E', the dual space of E,
and (yn)n>Ic F, as wellas a summablesequence (cn)n>Ic C suchthat
U(X)
Cn (Xn I X)yn
=E
n>1
for every x E E.
If E is a locally convex space and if p is a continuousseminormon E, set
= 0}.
Then Np is a closed subspace of E and p induces
Np = {x E E;p(x)
a norm on the quotient space Ep . Let Ep denote the completionof Ep . If q
is a continuousseminormon E satisfyingq > p, the identityon E induces a
continuouslinear mapping jq p from Eq to Ep. We say that E is nuclear if
foreverycontinuousseminormp on E, thereexistsa continuousseminorm
q > p such that jq,p is nuclear [Tr, Definition50.1].
3.1.7.
Theorem.
If F is a finitely generated group, the following properties are
equivalent:
(1)
F is of polynomial
(2)
H?? (F)
growth;
is contained in 1 (F);
(3) there exists p E ( 1,2)
such that H?(F)
(4) there exists c > 0 such that for every pair
the inclusion
(5)
H'(F)
Moreover,
of Hs(F)
in HS (F)
is contained
(s , s')
is of Hilbert-Schmidt
in lp(r);
which satisfies s'
<
s-c,
class;
is nuclear.
is fulfilled, then the inclusions
and one can take c = 2 (e'(F) + 1) in (4).
if one of the above conditions
and (3) are continuous,
in (2)
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
thereexist c and r > 0 such that ICkl
Proof. (1) =* (2) By hypothesis,
c(1?+k)r foreach k E N. Set s = ?+ r/2. If E Hs (F), one has
/
E
<1/2
=
gECk
1(
gECk
1/2
I l l
k>O
<
~~~~~~~1/2
E
k>O
?
k>O
gECk
k>O
187
2
g r+2
(g) I2+
gECk
C 11112,s
whichshowsthat Hs(F) is containedin 1l(F).
(2) = (3) is immediate.
thene'(F) = oo byLemma3.1.6.
growth
(3) => (1) If F is notofpolynomial
Consequently,
I = X foreveryk > .
su oglo
n>k
log n
sequence (nk)k>l
increasing
Thus,one can finda strictly
Let pe(I,2)
in N suchthat
foreveryk>1.
IC_ I>(nk)k
andset
=E
k>l
Cn
I/P"
iCnk
that i2Is is finiteforeverypositivereal numbers,
It is readilyverified
= ??.
butthat 11
V/11P
(1) =>. (4) If s and s' satisfys-s' > (e'(F)+ 1)/2, let c > 0 be smallenough
in orderthat
s - s' > (e'(F) + 1 + 8)/2.
Thenthereexistsa., > 0 suchthat
lCkl <?a,jl
+ k)e +g
foreveryk > 0.
function
Set J(s) = (1 + JgJ)SJg,for g e F, where6g is thecharacteristic
of {g}.
of Hs(F)
The family (J(S))
SF
g )gEr' is the canonical basis
If J is theinjectionof Hs (F) into Hs (F) one gets
2
,lJ( 5s)) 11 =s
11,2 S
gEr
< ag E(1 +
k>O
because 2(s' - s) + e'(F) + c < -l
E
k>O
k)2(s'-s)+e?+g
Ck
<
|( 1 +
k)2('s
PAUL JOLISSAINT
188
(4) => (5) If p is a continuousseminormon Hoo(F), one can findc and
s > 0 suchthatp((O) < cIIq'I2,, foreveryepe H' (F) . Sincethecomposition
mappingsis nuclear,using(4), thereexistss > s' such
of twoHilbert-Schmidt
thattheinclusionof Hs(F) in Hs (F) is nuclear.Thenthe mappinginduced
by theidentity
fromHs(F) to H??(F)p is nuclear.
(5) =- (1) If H??(F) is nuclear,thereis an s > 0 suchthattheinclusionJO
of Hs(() into HF(r) = 12(r) is nuclear.In particular,JOis Hilbert-Schmidt
and theseries
S
gEr
IIJo(g5s))II
=
IICkl(l +
k>O
k)-2s
whichshowsthat F is of polynomialgrowth. Q.E.D.
converges,
(RD) ifand onlyif F
3.1.8. Corollary.If F is amenablethenF has property
is ofpolynomial
growth.
(RD) by Theorem
Proof.If F is of polynomialgrowththenit has property
3.1.7(2).
If F is not of polynomialgrowth,thereexists V/E H' (F) whichtakes
nonnegative
values,suchthat V 0 1l'(F). As F is amenable,one has forevery
(
e
function 1'(F), with (0(g) > 0 foreveryg e F
' 119111'
_<llt(4)1 ' 11AWv11
11?11,
of F. It followsthat F cannothave
where t is the trivialrepresentation
property
(RD). Q.E.D.
3.1.9. Corollary.Consideran integern > 3. Then SL(n, Z) does notpossess
property
(RD).
Proof.Let a denotethe matrix( 2 l ) (whichbelongsto SL(2, Z) ) and let
F = ZX2X Z be the corresponding
semidirectproduct. F is embeddedin
with
thestandardsemidirect
productZ2 X SL(2, Z), whichmaybe identified
the subgroupof SL( 3, Z) constituted
by matricesof the form ( l), where
.
Thus
is
a
E
Z2
and
A
E SL(2, Z)
F
u
subgroupof SL(n, Z) for n > 3. By
to verify
that F does nothaveproperty
(RD). But
Proposition2.1.1, it suffices
F is solvableand of exponential
growth(cf.J.Tits,Appendixto [Gro,Lemma
(RD). Q.E.D.
3]). By theabove corollary,F does nothaveproperty
ofhyperbolic
isometries.In thisparagraph,X is
3.2. Discontinuous
subgroups
a completenoncompactRiemannianmanifoldwithboundedstrictly
negative
0
such that
sectionalcurvatureK. More precisely,thereexist K1 < K2 <
<
X
<
E
x
for
every
(cf.
[BGS,
K(x)
K2
K1]).
K,
For example, the hyperbolicspaces Hn (R), H n(C), Hn(K) and H 2(0) satisfytheaboveconditions[Mo, Chapter19] whereK is thefieldof quaternions
and where0 denotesCayleynumbers.
RAPIDLY DECREASING
FUNCTIONS IN REDUCED
C*-ALGEBRAS OF GROUPS
189
Now we describesimplyconnectedRiemannian spaces withconstantnegative
= {x E Rn
sectionalcurvature[KI, 1.11.8]: Let p > 0 and n > 2; set 97p
p
is a Riemannian space with sectional curvature K = p -2
lxl < p}. Then 97p
p
with respectto the metric
ds=
21dxl
1_
-XI2
Let us consider a geodesic trianglein 97p with angles a , f and y, let a
(resp. b , c ) be the lengthof the opposite side of a (resp. /B,y ). Then (a, /B,y)
and (a, b, c) satisfythe sine rule[KI, 2.7.5]
sina
sinh(a/p)
sin y
sinh(c/p)
sin ,B3
sinh(b/p)
Let F be a discontinuousgroup of isometriesof X and let 0 be a point of
in Lemma 1.1.4,
X. Set L(g) = d(O, g(O)) foreveryg E F. As it is remarked
L is a length-function
on F satisfyingfor every x E X
d(x, g(x)) - 2d(O,x) < L(g) < d(x, g(x)) + 2d(O,x).
Though L dependson thechosenpoint 0, thespaces HL(F) and HL (F)
do not.
Here is the main resultof the paragraph:
3.2.1. Theorem.If F is cocompact(i.e. X/F is compact),thenF hasproperty
thereexistsa positiveconstantc whichdependsonlyon
(RD). Moreprecisely,
theactionof F on X such that koH? c1koH22L foreveryfinitely
supported
ep
function on F.
Note that,by Theorem 6.15 of [Rag], such a groupis alwaysfinitely
presented.
Thus H? (F) is contained in C,*(F). When F is finitelygeneratedbut not
cocompact, we do not know whether H? (F) is contained in C,*(F) except in
dimension 2 (Corollary2.1.6). However, HL (F) is not contained in C* (F) in
m
general: a simple example is supplied with the action of F= M2(R){
m E Z} by homographictransformations
on the Poincare half-plane H
In
this
one
can
set
>
0}
.
C; Im(z)
case,
L(g) = d(i, g(i))
If g= (OIl
forg E F.
onehas
2log(1 + Iml) - 21og2 < L(g)
< 2log(1 + Iml)
since the distance between two points z and w in H2 is given by
I + Iz
d(z, w) = log ( z -'C
-w )
Note that F is of exponentialgrowthwith respectto L.
Set
if m<0,
tt1 mAA
0,
<' Jj0 1
m
if m > 0.
=
{z E
190
PAUL JOLISSAINT
One verifies
easilythat V belongsto HL (F), butthat
I 2q + 1 )2 >
forevery q > 1 . Thus V * V does not belongto
-2/3
2
(F). Finally, F is a subgroup
of SL(2, Z) whichimpliesthatHLj (SL(2, Z)) is nota convolution
algebraand
consequently
notcontainedin Cr,(SL(2, Z)) .
Let us nowgivetheproofof Theorem3.2.1. Fromnowon we supposethat
X/F is compactand,without
loss of generality,
thatthestabilizerof 0 in F is
trivial.Thereis thusa a > 0 suchthat
d (u(O), v(O)) > 25
foreach pairof distinctelementsu and v of F. Sincethelength-function
L
is henceforth
fixed,we write
Cr = {g e F;r-
and
Xr
1 <L(g)
< r},
thecharacteristic
function
of Cr. Set also
Cr,a
=
{g
E
;
r - a < L(g) < r + a},
if r and a belongto R+ .
3.2.2. Lemma.Let c ,k, 1 and p be nonnegative
real numbers
suchthatp <
Consider
in
X
a
whose
verticesx, y and z satisfy
triangle
min(k, 1).
d(x,z)=k
d(x,y)=k?+-2p-c,
and d(y,z)=l.
Let x' denotetheuniquepointon thegeodesicsegment[x,y] whichsatisfies
d(x,x') = k -p.
Thenthereexistsa > 0 independent
of k, 1 and p suchthatp < d(x', z) <
p+ a.
Proof.The sectionalcurvatureK of X satisfiesK < -p2 forsomepositive
p. Considerthena trianglein ign withverticesxp, yp, zp, and let xp E
properties:
[xp, yp] withthefollowing
( 1) d(xp ,) = k +1- 2p -c, d(xp ,x= k - p and d(x' , z)= d(x', z);
(2) theangle 0 at x'P between[x',xp]
, z] is equal to theangle
P p and [x'PP
at x' between [x', x] and [x', z].
We adoptthefollowing
notations:
k' = d (xp, z ),
I'
d (yp,zp)
r = d (x', z) =d (x' , zp)
ce is the angleat xp between[xp,yp] and [xp,zp]; ,B is the angleat yp
between[y ,xp] and [yp zP]; y, and Y2 aretheanglesat zp between[zp, xP]
and [zp,x ] and between[z ,x ] and [z ,yp] respectively.
FUNCTIONS
RAPIDLY DECREASING
IN REDUCED
C*-ALGEBRAS OF GROUPS
191
/
x
y
xp
FIGURE
one gets
By the sine rule in Sp
sinh
r
hzX
sinca
tp)
sin 0
sin
1
hk'0
sinh -JJ
sin a
sin(y1+
p)
Y2)
sin(y1+ Y2) . ksinh(k
sin 0
p
cO sinY2
siny O5
=C(OiS
+ COSyi, )
C?Y2
sin 0
sin0
sin(k'~p sinh(I'~
i(P
sinh(k+-2p-c)
sinh((k - p)/p) sinh(l'/p) + sinh(k'/p) sinh((l - p - c)/p))
sinh((k + I- 2p - c)/p)
since
siny,
sin 0
sinh((k-P)/P)
sin(k'/p)
and
sin_
sin 0
-
sinh((l-p -c)/p)
sinh(l'/p)
By comparison of the sinh functionwiththe exponentialone, one has
erlP <
c epIp
because the inequality K < -p2 implies k' < k and 1' < I [KI, 2.7.6]. Thus,
r < p +a wherea > 0 and independent
of k , I and p . Moreover,r+k -p > k
and then r > p.
Q.E.D.
3.2.3. Lemma.Let a and b be positiveconstants.Thereexistsa positive
numberN, dependingonlyon a, b and on theactionof F on X, withthe
realnumbersk and I andfor
following
property:
For everypair ofnonnegative
each g E Ck+/, one has
l{h EECka;h
lg E Clb}| < N.
Proof.Let (g, h) E Ck+/ X Ck ,a be such that h 1g E C1b . Let us considerthe
geodesic trianglein X withvertices 0, g(0) and h(0) . If x E X is the unique
point on [0, g(0)] such ihat d (0, x) = k, Lemma 3.2.2 (with p = 0 ) ensures
192
PAUL JOLISSAINT
the existenceof a positiveconstantA, independent
of k , I, g and h, such
that d(x , h(O)) < A. This impliesthattheball B(h(O) , 6) withcenterh(O)
and radius 3 is containedin B(x ,A + 6) . Sincethedifferent
balls B(h(O) ,c)
are disjoint,and sincethesectionalcurvature
is boundedbelow,thenumberof
h's in Ck,a suchthat h g E C/,b is boundedby a constantindependent
of
k,l and g. Q.E.D.
Now we exposethemaintechnicalresultfortheproofof Theorem3.2.1 (by
Proposition1.2.6); it is a generalization
of Lemma 1.3 of [Haa]:
3.2.4. Proposition.Thereexistsa positiveconstantc, dependingonlyon the
actionof F on X, withthefollowing
property:
If k, 1,m E N satisfyik- 11<
qi
<
m k + 1, if (o, E CF are supported
in Ck and C1 respectively,
then
II(Q * '/)xmII2< CIIPI1211k/112.
Proof.We distinguish
twocases: (i) m = k + 1, and (ii) Ik - 11< m < k + 1.
Let us firstshowthefollowing
moregeneralassertion:
(A) If a and b are positiveconstants,thereexistsa positivenumberc
dependingonlyon a , b and on the actionof F on X, withthe following
property:If k', I' E R+, if p', yv E CF are supportedin Ck ,a and C1,b
respectively,
then
11(0P* V")XkI+l'112< C
112112 112*
In fact,usingLemma 3.2.3, thereexists N > 0 such thatforeveryg E
Ck,+1,,the numberof h's in Ck, a suchthat h g E C1,b is boundedby N.
Then,usingtheCauchy-Schwarz
we get
inequality,
ZE
ko' * q"(g)I < N
(hi,h2)ECk
I?'(h)121
ql'(h2)12
,aXCl/ ,b
h1h2=g
hence
11(9'* V)Xk,+1ll
2< N
,{
gECk,+i,
< N
v12
E
\
1?
l'(h,)l21 V'(h2)12
(h ,h2)ECk1,aXCl/,b
hlh2ECk,+I,
11W12
Thus,Case (i) is a directconsequenceof (A), witha = b = 1 .
Suppose now that Ik - lI < m < k + l; then m = k + I - 2p for some
half-integer
p < min(k, 1) . Let g E Cm.
Let x'(g) be the unique point of [0, g(O)] such that d(O , x'(g)) = k - p.
Since X/F is compact,thereexistsr > 0 suchthatthedistancefromx'(g) to
theorbitof 0 is at mostequal to r. Therethenexistsu E F whichsatisfies
RAPIDLY DECREASING FUNCTIONS IN REDUCED C*-ALGEBRASOF GROUPS
193
\
(0)
Q~~~~~~ug
/
FIGURE
2
d(ug(O),x'(g)) < r. Set vg = ui g. Then g is writtenin "reducedform"
inequalitieshold:
g = Ugvgsincethefollowing
k - p - r < L(ug) < k - p - r
and
l-p-r-l<L(vg)<l-p+r.
Thanksto Lemma3.2.2,if (hIlih E Ck x C, and h1h2= g, then
h
a < L(u1ghi)
p+a
wherea > 0 dependsonlyon the actionof F on X. Then we getthatif
g E Cm can be writtenas g = hIh2 with (hI,h2) E Ck x C1, thereexists
necessarilyw E Cp a suchthat h1 = ugw and h2 = wuYvg. Set
p
-
1/2
4p(U) =
if k -p - r < L(u) < k-p + r,
)12
CpUa
otherwise,
0
and
1/2
(op ()
%(V){(W~~'u)E
l(w-v)I2
Cp,a)
if I -p-r < L(v) < l -p + r,
otherwise.
O
One has
2
WECp,a
gECm
<
1:
(
gEC,n
=I(p *
(U )2tV/ (Vg)2
,(?9q
* W (g))2
gECm
< N(a,r)2N(a,r+
tI/p)XmI2
1)II1II2lIItuI2
194
PAUL JOLISSAINT
usingAssertion(A), whereN(a, r) and N(a, r + 1) comefromLemma3.2.3.
Q.E.D.
Finally,usingthe same arguments
as in Lemma 4 of [Fl, p. 213], we can
adaptour proofof Theorem3.2.1 and get
3.2.5. Proposition.
If F is a geometrically
finiteKleiniangroupwithout
parabolic elements,thenthereexistsa positiveconstantc dependingonlyon the
actionof F on H3(R) suchthat11H1< C11P12,2,L forevery
supported
finitely
function(o on F.
APPENDIX.
PROPERTY
(RD)
FOR LOCALLY COMPACT GROUPS
Let G be a locallycompactsecond countablegroup. Let CC(G) denote
thespace of compactlysupportedcontinuousfunctions
on G and L2c (G) the
spaceof (classesof) measurablefunctions
whichare square-summable
on every
compactsubsetof G.
If s is a real numberand if L is a continuouslength-function
on G, the
space Hs(G) is theset of (classesof) functions(oE L2o(G) suchthat
11k12,s,L
=
I(
(g)
(I + L(g))
dg
is finite.
The space of rapidlydecreasing
on G (withrespectto L) is the
functions
intersection
of the spaces HL(G). We denoteit by HL (G). It is a Frechet
space forthetopologyinducedbythefamilyof norms(11 2 s,L)sER
A.1. Definition.G is said to haveproperty
(RD) if thereexista continuous
< c11o12,s,Lfor
length-functionL on G and c and s > 0 such that 11H(q)11
everyfunctionq' E CC(G).
A.2. Proposition.If G has property
(RD) and if L is a continuouslengthA.1, thenHLJO
functionon G whichsatisfiestheconditions
ofDefinition
(G) is
an involutive
and involution)
algebra(withrespectto naturalconvolution
which
is identified
witha densesubalgebraof C, (G).
bytheleftregularrepresentation
The proofof PropositionA.2 is thesame as thatof Lemma 1.2.4.
We are goingto show
A.3. Proposition.Let G be a secondcountablelocallycompactunimodular
groupand let L be a continuous
on G. SupposethatG contains
length-function
F
a discrete
such
that
in C,*(F), where
cocompact
subgroup
HLO(F) is contained
of L to F. Then G has property
(RD).
Lo is therestriction
A.4. Corollary.If G is a connected
Lie groupofreal
noncompact
semisimple
rankoneandfinite
thenG possessesproperty
center,
(RD). (Comparewith[He].)
Proofof Corollary.Let K be a maximalcompactsubgroupof G and L a
K-invariant
on G comingfroma Riemannianmetricon the
length-function
195
RAPIDLY DECREASING FUNCTIONS IN REDUCED C*-ALGEBRASOF GROUPS
homogeneous space G/K, i.e. L(g) = d(xo, g(xo)) for every g E G, where
x E G/K. If F is a cocompactsubgroupof G then HLO(F) is containedin
C*(F) byTheorem3.2.1. Q.E.D.
A.3. If X = F\G denotesthe set of rightcosetsof G
Proofof Proposition
measure
probability
moduloF thenX is compactand possessesa G-invariant
,u. By Theorem8.11 of [Var],thereexistsa regularBorelsection a for X;
moreprecisely,-a is a BorelmappingfromX to G satisfying
(1) 7ro a = idx, wherei is thecanonicalprojectionof G onto X;
compact.
(2) u(X) is relatively
Notethatthemap -* fX(EEr, (ya(x))) du(x), definedon C,(G), is a Haar
measureon G.
thereexist c and s > 0 suchthat
By hypothesis,
?C IIf
11f ,s,LO
IIf2112
Ifl* f2112
forall f, f2E Cr
Let (o, qi be elementsof CC(G). We have
l
l
?
l| (i EF {Z
13EV
JXX
ff
Zxx
=
1
zE Ix IE(S5Y)/(fy
=x
Y))
du
d,iCv)d/t(x)
kH2,s,Lo 1I,xH2 ddi(y) d8Xx
where q,,(y) = fo(yo(y)) and Wy,x(Y)= W(yY1 yo(x))
Thuswe get
2C
?
sup(EX(l+ L(a(x)))s
and L is continuous. Q.E.D.
where M=
foreveryy e F.
Ix
dul(jy)
H2du(x))
HvI/y(x
X1f} 12, (Ix
<c2 lM/ f2l(z
22
c2
d(x)
Go1(,?)vI/
,xj12d,i(y)
d,i(x)
< c l|ix
2( )1
d/i(y)2) djt(x)
(y)) V(a(0)2)
0Y(fl)l'Y,x(IY
lXlx l
5x
L
()2(1)
dud(y)
(l+L(y))2
)
2dydx
compact
is finitesince (X) isrelatively
theconverseof PropositionA.3 holds.
We do notknowwhether
196
PAUL JOLISSAINT
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RUE DU LIEVRE 2-4, CASE POSTALE
UNIVERSITEDE GENEVE, SECTION DE MATHEMATIQUES,
240, CH-1211 GENEVE24, SWITZERLAND
Current
address:Universitede Neuchatel, Institutde Mathematiques, Chantemerle20, 2000
Neuchatel,Switzerland