Kazhdan projections, random walks and ergodic theorems

0.39 MB·English

by  Cornelia Druţu

KAZHDAN PROJECTIONS, RANDOM WALKS AND ERGODIC
THEOREMS
5
1
0 CORNELIADRUT¸UANDPIOTRW.NOWAK
2

t
c
O
ABSTRACT. InthispaperweinvestigategeneralizationsofKazhdan’sprop-
1 erty(T)tothesettingofuniformlyconvexBanachspaces. Weexplainthe
1
interplaybetweentheexistenceofspectralgapsandthatofKazhdanpro-

jections. Our methods employ Markov operators associated to a random
]
R walkonthegroup,forwhichweprovidenewnormestimatesandconver-
gence results. This constructionexhibitsuseful propertiesand flexibility,
G
and allows to reclassify the existence of Kazhdan projections from a C -
∗
.
h algebraicphenomenon to a natural objectassociatedto randomwalkson
t groups.
a
We give a number of applications of these results. In particular, we
m
addressseveralopenquestions. Wegiveadirectcomparisonofproperties
[ (TE) and FE withLafforgue’s reinforcedBanachproperty(T); we obtain

shrinking target theorems for orbits of Kazhdan groups and apply them
3
v toaquestionofKleinbockandMargulis;finally,weconstructnon-compact
3 ghost projections for warped cones, answering a question of Willett and
7 Yu. In this last case we conjecture that such warped cones provide new
4 counterexamplestothecoarseBaum-Connesconjecture.
3
0
.
1 1. INTRODUCTION
0
5 Onewaytoinvestigatepropertiesofgroups,especiallywithaviewtotheir
1 actions on Banach spaces, is through the group Banach algebras. These are
:
v naturalanalyticobjectsencodingmanypropertiesofthegroup. Theexistence
i of projections in such algebras is a particularly important and challenging
X
problem. For instance, the(non)existenceof idempotentsother than 0 and1
r
a in the reduced group C -algebra of a torsion-free group is a long-standing
∗
conjecture of Kadison and Kaplansky. When the group is amenable (and
more generally, a-T-menable) the Kadison-Kaplansky conjecture is known
to be true. Additionally, for amenable and torsion free groups the maximal
group C -algebra is isomorphic to the reduced group C -algebra, therefore
∗ ∗
themaximalgroupC -algebradoesnothavenon-trivialidempotentseither.
∗
Date:October13,2015.
TheresearchofbothauthorswassupportedbytheEPSRCgrant“Geometricandanalytic
aspectsofinfinite groups". The researchofthe firstauthor wasalsopartiallysupportedby
theprojectANRBlancANR-10-BLAN0116,acronymGGAA,andbytheLabexCEMPI(ANR-
11-LABX-0007-01). TheresearchofthesecondauthorwaspartiallysupportedbyNarodowe
CentrumNaukigrant2013/10/EST1/00352.
1
2 CORNELIADRUT¸UANDPIOTRW.NOWAK
A main result of this paper is an explicit construction of proper idempo-
tents in many group Banach algebras. The constructionis based on random
walks,anewingredientinthissetting. Ourconstructionturnsout toberel-
evantinvariouscontexts,fromexpandergraphstoergodicgeometryandthe
Baum-Connesconjecture.
A Kazhdan projection for a locally compact group G is an idempotent in
themaximalgroupC -algebraC (G),whoseimageunderanyunitaryrep-
∗ m∗ax
resentation is the projection onto the space of invariant vectors. Such pro-
jections exist in C (G) if and only if the group G has Kazhdan’s property
m∗ax
(T)[1]. Theyareimportantformanyapplications. Aclassicalconsequenceof
their existence is the fact that the map on K-theory induced by the canoni-
calhomomorphismC (G) C (G)fromthemaximaltothereducedgroup
m∗ax → ∗r
C -algebra, failsto bean isomorphismfor Kazhdangroups, see e.g. [18, Ch.
∗
2,S.4]. TheyplaythemainroleinthefailureofsomeversionsoftheBaum-
Connes conjecture, since projections of this type cannot be in the image of
theBaum-Connes assembly map [27]. Kazhdanprojectionsarethe main in-
gredient of Lafforgue’s reinforced Banach property (T) and allowed for the
constructionofthefirstexamplesofexpanderswithnocoarseembeddinginto
anyuniformlyconvexBanachspace[35]. Finally,theyalsoplayanimportant
roleinthegeneralizationofproperty(T)toC -algebras[11].
∗
However,Kazhdanprojectionshavealwaysbeenconsideredsomewhatmys-
teriousobjects, whose existencecan eventually be ascertained,but whocan-
notbeconstructedexplicitly(seee.g. [28,Footnote22,page73]).
Spectral gaps, Markov operators and projections. At the core of our
paper is a study of a Banach space version of property (T), formulated in a
very general setting: with respect to a given family of isometric representa-
tionson Banach spaces. We provethat such a propertycan be characterized
in three different ways: the standard spectral gap property, the behavior of
theMarkovoperatoronacanonicalcomplementofthefixedvectorssubspace,
andtheexistenceofaKazhdanprojection,withan explicitformulatocalcu-
lateit,usingMarkovoperators.
Indeed,givenanisometricrepresentationπofagroupG onareflexiveBa-
nachspaceE,thesubspaceEπoffixedvectorshasacanonicalπ-complement,
µ
E (see Section 2.c for details). Given a probability measure µ on G, let A
π π
denote the Markov (averaging) operator associated to π via µ. We prove the
following.
Theorem1.1. LetG bealocallycompactgroup,andF afamilyofisometric
representations of G on a uniformly convex family E of Banach spaces. The
followingconditionsareequivalent:
(i) thefamilyF hasaspectralgap(seeDefinition2.3);
(ii) there exists a compactly supported probability measure µ on G and
λ 1suchthatforeveryisometricrepresentationπ F ofG on E E
< µ ∈ ∈
wehave Aπ|Eπ <λ;
° °
° °
KAZHDANPROJECTIONS,RANDOMWALKSANDERGODICTHEOREMS 3
(iii) thereexistsacompactlysupportedprobabilitymeasureµonG anda
number S such that for every π F the iterated Markov opera-
<∞ ∈
tors Aµ k converge with speedsummable to at most S to the projec-
π
tionP ontoEπ alongE ,thatis
¡ π ¢ π
Aµ k P a ,
π π k
− ≤
° °
where a S. °¡ ¢ °
k k≤ ° °
In TheoremP3.8we give an explicitformulafor theprojectionP interms
π
oftheNeumannseriesoftheMarkovoperator:
(1) P I ∞ Aµ n I Aµ .
π E π E π
= −Ãn 0 ! −
X= ¡ ¢ ¡ ¢
The hypothesis of uniform convexity is needed only in the implication (i)
(ii) and (iii), for the other implications it suffices to have a family of com-
⇒
plementedrepresentationsonBanachspaces,inthesenseofDefinition2.2.
When G has Kazhdan’s property (T) and E is a Hilbert space, Theorem
1.1 holds for F the family of all unitary representations of G. However, as
Theorem 1.1 is formulated in terms of a family of representations, it also
applies in the setting of Property (τ) (see Section 5 and the corresponding
paragraphlaterintheIntroduction),ofproperty(Tℓp)introducedin[5]etc.
TheequivalenceinTheorem1.1hasaneffectivesidetoit,describedbelow.
Given a Kazhdan pair (Q,κ) defining the spectral gap (see Definition 2.3),
the conditions (ii) and (iii) hold for a large class of measures, which we call
admissiblewithrespecttotheKazhdansetQ,explicitlyconstructedbymeans
ofQ,seeDefinition2.1. For everysuchmeasureµ,aconstantλasin(ii)can
becomputedusingtheKazhdanconstantκ,themodulusofuniformconvexity
ofthefamilyE andthechoiceofanappropriatecompactlysupportedfunction
on G associated to µ. Property (iii) then holds with a λk. Conversely,
k
=
given a measure µ and λ (0,1) satisfying either (ii) or (iii) with a λk,
k
∈ =
the support of µ is a Kazhdan set and its corresponding Kazhdan constant
is 1 λ. This implication applies, for instance, in the case of semisimple
−
Lie groups with finite center, and their unitary representations on Banach
spaces, to anyprobabilitymeasurewith symmetric support not containedin
aclosedamenablesubgroup[59,TheoremC].
One of the advantages of Theorem 1.1 is the high degree of flexibility in
ensuring uniformity of several parameters for classes of isometric represen-
tations. Forinstance,inthecaseofgroupsadmittingfiniteKazhdansets(see
Section3.g)thisuniformitydependsonlyonthreequantities:
(a) theKazhdanconstantofthefamilyofrepresentations,
(b) thecardinalityoftheKazhdansetQ,
(c) themodulusofuniformconvexityoftheBanachspaces.
ItdoesnotevendependonthegroupG,aslongaswecanarrangetheabove
threeitemstohaveuniformbounds.
4 CORNELIADRUT¸UANDPIOTRW.NOWAK
For applications,theexistenceof finiteKazhdansets isaconsiderableas-
set: the averages become finite, the random walks discrete and an algorith-
mical approach and the use of computer become possible (see for instance
Theorem 3.8, Remark 3.10 and Section 6). As it turns out, the existence of
such finite sets is ensured in many cases outside the class of finitely gener-
atedgroups,andinmanycasesthesetsaredescribedexplicitly,asexplained
brieflyinSection3.g.
KazhdanprojectionsingroupBanachalgebrasandLafforgue’srein-
forced Banach property (T). The uniform convergence described in The-
orem 1.1, (iii), depending on the Kazhdan constant, the modulus of uniform
convexity of the family E, and the choice of the measure µ, shows that the
existenceofaKazhdanprojectioningroupBanachalgebrasisaconsequence
of a uniform version of property (TE). Property (TE) was introduced and
studied in [21,2] as a natural generalization of property (T) from Hilbert to
Banachspaces.
Theorem1.2(seeTheorem4.6andCorollary4.7). LetGbealocallycompact
group and let F be a family of isometric representationsof G on a uniformly
convexfamilyofBanachspaces. ThereexistsaKazhdanprojection p CF(G)
∈
ifandonlyifthefamilyF hasaspectralgap.
In particular,ifG hasKazhdan’sproperty(T)thenthereexistsa Kazhdan
p
projectioninthe L -maximalgroupalgebraC (G)forevery1 p .
p max
< <∞
Here, CF(G) is a natural version of the maximal C∗-algebra of G for the
family F of representations, see Definition 4.1. Banach group algebras for
largerthanisometricclassesofrepresentationswereintroducedandstudied
by V. Lafforgue [35]. We point out that Theorem 1.2 gives an entirely new
proof of the existence of a Kazhdan projection even in the classical case of
property (T) and Hilbert spaces. Only two previous proofs are known. The
firstisduetoAkemann andWalter[1]anditreliesonpositivedefinitefunc-
tions,atoolavailableessentiallyonlyinHilbertspaces. Anotherproof,using
minimal projectionsin C -algebras, is due to Valette [63]. The topic of oper-
∗
ator algebras on L -spaces isan emerging directionin non-commutative ge-
p
ometry. Additionally,an approachto theNovikov conjecturevia L -versions
p
oftheBaum-ConnesconjecturehasbeenrecentlydevelopedbyKasparovand
Yu. Theorem 1.2 shows that for groups with property (T) the same obstruc-
tionsasintheHilbertspacecasearelikelytoexistin K-theory.
Theorem 1.1 allows to compare V. Lafforgue’s definition of reinforced Ba-
nach property (T) [35] to other generalizations of property (T) to Banach
spaces,i.e. properties(TE)andFE[21,2]. Thequestionofsuchacomparison
hasbeen consideredbyseveral expertspreviously. In [36]it wasshown that
thereinforcedBanachproperty(T)implies FE,andin[2]it wasshownthat
property FE implies(TE). SinceLafforgue’s reinforcedBanachproperty(T)
is formulated in terms of existence of Kazhdan projections in certain group
Banach algebras, Theorem 1.2 provides implications in the other direction.
WediscussthisindetailinSection4.b.
KAZHDANPROJECTIONS,RANDOMWALKSANDERGODICTHEOREMS 5
Property (τ) and expanders in the Banach setting. As Theorem 1.1
holdsforafamilyofrepresentations,itcanbeappliedinthecontextofprop-
erty (τ), introduced by Lubotzky. Thus, we use Theorem 1.1 to formulate a
complete, canonical generalization of property (τ) to uniformly convex Ba-
nach spaces (This contrasts with the situation for property (T), for which
thereareseveralcompetinggeneralizations). Ourgeneralizedproperty(τ)is
moreover consistent with a notion of expanders for Banach spaces, defined
usingPoincaréinequalities(seeDefinition5.2).
Moreprecisely,forauniformlyconvexBanachspaceE weintroduceprop-
erty (τE) by the same definition as for Hilbert spaces, requiringthat certain
isometric representations factoring through finite quotients of G are sepa-
rated from the trivial representation; that is, they have a uniform spectral
gap. ThefollowingisaconsequenceofTheorem1.1.
Theorem 1.3. Let E be a uniformly convex Banach space, let G be a finitely
generatedresiduallyfinitegroupandletN {N }beacollectionoffiniteindex
i
=
subgroupswithtrivialintersection. Thefollowingconditionsareequivalent:
(i) G hasproperty(τE)withrespecttoN {Ni} andasymmetricKazh-
=
dansetQ;
(ii) theCayleygraphsCay(G/N ,Q)formasequenceof E-expanders;
i
(iii) thereexistsaKazhdanprojection p CN(E)(G).
∈
In the Hilbert space case the algebra appearing in the condition (iii) is a
C -algebra. Note that the Kazhdan set Q in Theorem 1.3 does not neces-
∗
sarily generate G. Examples of Kazhdan sets that are not generating exist
alreadyforgroupsG andcollectionsN havingtheclassicalproperty(τ). For
instance,ifG SL (Z),afinitesymmetricsetgeneratingasubgroupZariski
n
=
densein SL (R)isaKazhdansetforanappropriatechoiceofN (see[9]and
n
references therein). See also [58] for an earlier example of non-generating
Kazhdansetforactionsonexpandersthatarefinitequotients.
Applicationstoergodictheory. AnotherareainwhichTheorem1.1finds
natural applications is ergodic theory. Consider, for instance, a group with
property (T) acting ergodically on a probability space (X,ν), and let f be an
arbitrary function in L2(X). If the two operators appearing in the equality
(1) are applied to f, then the left hand side becomes fdν, and the entire
X
formulabecomesaBirkhoff-typetheorem,inwhichanexactexplicitformula
R
isprovided,insteadofjustanestimatefortheremainder.
Thus,astrikingconsequenceofourresultsisthat,whileforergodicactions
of amenable groups the best way to average is via sequences of Følner sets,
for ergodic actions of groups with property (T) a most effective averaging is
viasequencesofmeasureswithcompactsupportapproximatingtheKazhdan
projection.
Generalizations to group actions of ergodic theorems of Birkhoff and von
Neumannhavebeenanobjectofsignificantinterest,see[12,43]forasurvey
andhistoryofthissubject. Anearlyresultofthistypeistheclassicalergodic
6 CORNELIADRUT¸UANDPIOTRW.NOWAK
theorem of Oseledec [50], that the time averages of a function over convolu-
tionpowersofaprobabilitymeasureconvergetothethemeanofthefunction
over the space. Theorem 1.1 allows to achieve a much stronger type of con-
vergence, in norm topology instead of the strongoperator topology, and with
uniform estimates, depending on the spectral gap. It also allows to average
usingmeasureswithfinitesupport(equaltoaKazhdanset)eveninthecase
ofnon-discretegroups(seeSection3.g).
Theorem 1.4. Let G be a locally compact group and let (Xi,νi),i I, be a
∈
familyofprobabilityspacesendowedwithmeasurepreservingergodicactions
of G. Consider also a collection E of uniformly convex Banach spaces, and a
number p in(1, ).
∞
Assume that a family F of isometric representationsof G on Lp(Xi,ν;E),
with i I and E E, induced by the measure preserving actions of G, has a
∈ ∈
spectralgapandlet(Q,κ)beaKazhdanpair.
For everyQ–admissiblemeasureµ on G thereexists λ 1, dependingonly
<
on p, the normalizing factor of µ, the modulus of convexity of E, and κ, such
that
(2) Aµπkf f dν λk f p.
− ≤ k k
° ZX °p
° °
pPrreocoifs.eSlyi,nfcreomPπTh=eoXrefmd°°ν3,.4t)h.e assertion°°follows from Theorem 1.1 (or, mor(cid:3)e
R
Theorem 1.4 becomes very concrete in certain cases. When G has prop-
erty(T) and E R, weobtain auniformquantitativeergodic theoremfor all
=
probabilitypreservingactionsofG andforanyfixed1 p , seeTheorem
< <∞
6.2. Another case is when G SL (F) for F a non-archimedean local field.
3
=
It follows from [35] that for any uniformly convex E and any 1 p the
< <∞
family of isometric representations of G on Lp(X,ν;E) has a spectral gap.
Consequently, the above theorem holdsfor such G and any fixed E and p as
above.
AnothertypeofproblemstowhichTheorem1.1canbeappliedareshrink-
ingtargetproblems,whichaskhow oftendoesatypicalorbitofanactionhit
a sequence of shrinking subsets. This problem for orbits of cyclic (or unipa-
rameter) groups in locallysymmetric spaces, and for shrinkingsequences of
neighborhoodsof a cusp, has been thoroughly investigated and answered by
Sullivan [62] and Kleinbock and Margulis [32]. The latter formulated the
problem of finding similar results for shrinking sequences of neighborhoods
of a point. Although some estimates are known in the case of rank 1 locally
symmetricspaces,theproblemiscompletelyopeninthecaseofhigherrank.
Theorem 1.1 allows to provide quantitative estimates in terms of random
walksforthebehavior ofanergodicactionofagroupwithproperty(T)with
respect to a shrinking target. For instance, we have the following theorem
(we refer to Section 6.b for details, stronger statements and other corollar-
ies).
KAZHDANPROJECTIONS,RANDOMWALKSANDERGODICTHEOREMS 7
Theorem 1.5 (see Theorem 6.6). Let G be a locally compact group, and Γ a
latticeinit. Let{Ω }beasequenceofmeasurablesubsetsinG/Γ.
n
Assume that a locally compact group Λ with property (T) acts ergodically
on G/Γ. Let µ be a probabilitymeasureon Λ admissible with respectto some
Kazhdanset,andlet X betherandomvariablerepresentingthe n-thstepof
n
therandomwalkdefinedbyµ.
(i) If nν(Ωn)isfinitethenforalmostevery x G/Γ
∈
P P(Xn(x) Ωn) .
∈ <∞
n N
X∈
(ii) If nν(Ωn)isinfinitethenforeveryε 0andforalmosteveryx G/Γ,
> ∈
(3) P P(Xn(x)∈Ωn)=SN+O SεN ,
n N
X≤ ¡ ¢
where SN n Nν(Ωn). In particular, P(Xn(x) Ωn) 0 for infin-
itelymany=n N≤. ∈ >
P∈
Moreover, when Λ is endowed with a word metric distΛ corresponding to
an arbitrary compact generating, in the theorem above one may obtain an
estimatesimilartotheonein(3)forthesmallerprobabilities
P(Xn(x) Ωn,distΛ(Xn,e) an),
∈ ≥
wherea 0isaconstantdependingonthechoiceofthewordmetricandofµ
>
(seeCorollary6.8).
TheseresultsapplyforinstancewhenG isasemisimplegroup,Γalattice
inG andΛaninfinitesubgroupofG,orwhenG/Γisthen-dimensionaltorus
andΛisasubgroupof SL (Z).
n
Obstructions to the coarse Baum-Connes conjecture. The final appli-
cation we present concerns obstructions to the coarse Baum-Connes conjec-
ture. In [26,27] it was shown that the coarse Baum-Connes conjecture fails
for coarse disjoint unions of expander graphs arising from an exact group
with property (T). The reason is the existence of a certain Kazhdan-type
projection, a non-compact ghost projection, which is a limit of finite propa-
gationoperators. Untilnow suchghost projectionswereconstructedonlyfor
expandersasabove. WillettandYuinformulatedthefollowing
Problem1.6(Problem5.4,[64]). Findothergeometricexamplesofghostpro-
jections.
Hereweprovideananswerbyconstructingnon-compactghostprojections
forwarpedcones[55].
Let G be a finitely generated group acting ergodically by probability pre-
servingLipschitzhomeomorphismsonacompactmetricprobabilitymeasure
space (M,dist,m). Assume that the measure m is upper uniform, i.e. it is
distributed uniformly over M with respect to the metric, see Definition 7.1.
DenotebyO O (M)thewarpedconeassociatedtotheactionofG on M,as
G
=
definedinSection7.a(seealso[55]).
8 CORNELIADRUT¸UANDPIOTRW.NOWAK
Theorem 1.7 (see Theorem 7.6). If the action of G on (M,m) has a spectral
gapthenthereexistsanon-compactghostprojectionG B(L (O))whichis a
2
∈
limitoffinitepropagationoperators.
Wealsoconjecturethatsuchwarpedconeswithnon-compactghostprojec-
tionsasprovidedbyTheorem1.7yieldanewclassofcounterexamplestothe
coarseBaum-Connesconjecture.
Acknowledgments. ThesecondauthorwouldliketothanktheMathemat-
icalInstituteattheUniversityofOxfordforitshospitalityduringa4-month
stay which made thiswork possible. Both authors thankMikael de la Salle,
Adam Skalski, Alain Valette, Rufus Willett and Guoliang Yu for valuable
comments.
CONTENTS
1. Introduction 1
2. Preliminaries 8
3. Randomwalks,projectionsandspectralgaps 11
4. KazhdanprojectionsinBanachalgebras 21
5. Expandersandproperty(τ)forBanachspaces 25
6. Ergodictheorems 26
7. Ghostprojectionsforwarpedcones 32
References 38
2. PRELIMINARIES
2.a. Uniform convexity. Let (E, ) be a reflexive Banach space, B(E)
E
k·k
the algebra of bounded linear operators on E and I(E) the group of linear
isometries of E. The modulus of convexity of E is the function δE :[0,2]
→
[0,1]definedby
v w
δE(t) inf 1 + ; v w 1, v w t .
= − 2 k k=k k= k − k≥
n ° ° o
TheBanachspaceE issaid°tobeu°niformlyconvexifδE(t) 0forevery t 0.
° ° > >
AfamilyE ={Ei}i∈I ofBanachspacesisuniformlyconvexifδE(t)=infi∈IδEi(t)>
0foreveryt 0. ThefunctionδE iscalledthemodulusofconvexityofthefam-
>
ilyE.
2.b. Admissible measures. Compactly supported probability measures on
topological groups and the corresponding random walks are central objects
inourarguments. Weintroducesomenotationandseveralstandingassump-
tionsonsuchmeasures.
ConsiderG a locallycompactgroup, endowed with a(left invariant)Haar
measureη. Foranyfunction f :G Cwedenoteγ f(g) f(γ 1g),γ,g G.
−
→ · = ∈
We consider two particular cases, before introducing the notion of admis-
siblemeasureinfullgenerality. LetQ beacompactsubsetofG.
KAZHDANPROJECTIONS,RANDOMWALKSANDERGODICTHEOREMS 9
Continuous admissible measures. Let α,β:G [0, ) be continuous func-
→ ∞
tionswithcompactsupportsatisfying
αdη 1 and β(g) s α(g), s Q,g G.
= ≥ · ∀ ∈ ∈
Z
Anothercontinuousfunction,whosecompactsupportcontainsQ,canthen
bedefinedbytheformula
α β
(4) ρ + , where M(α,β) (α β)dη.
= M(α,β) = G +
Z
Wecalladecompositionasin(4)an(α,β)-decompositionofρ.
ThefunctionρgivesrisetoaprobabilitymeasureµonGdefinedbysetting
dµ ρdη.
=
Discrete admissible measures. Now consider functions with finite support
α,β:G [0, ). With the above conditions formulated for such α and β,
→ ∞
we define ρ as in (4), where M(α,β) α(g) β(g) , and the
g suppα suppβ
= ∈ ∪ +
formula(4)isagaincalledan(α,β)-decompositionofρ.
P £ ¤
As ρ has finite support and ρ(g) 1, it gives rise to a purely
g suppρ
∈ =
atomicprobabilitymeasureonG.
P
Definition 2.1. A measure µc (respectively µd) will be called a continuous
admissiblemeasurewithrespecttoQ (respectively,discreteadmissiblemea-
surewithrespecttoQ)ifitisdefinedbya continuousdensityρ (respectively,
afinitelysupportedfunctionρ)admittingan(α,β)-decomposition.
A probability measure µ on G will be called admissible with respect to
Q if there exists t [0,1] such that µ tµc (1 t)µd, where µc and µd are,
∈ = + −
respectively,continuousanddiscreteadmissiblemeasureswithrespecttoQ.
The normalizing factor of the function ρ and its associated continuous or
discretemeasureµistheinfimumofM(α,β),takenoverall(α,β)-decompositions
ofρ,withQ fixed. Thisfactorwillbedenotedeither M or M ,dependingon
ρ µ
theobjectreferredto.
The normalizing factor of an admissible measure µ tµc (1 t)µd is the
= + −
number
t 1 t 1
−
M − .
µ
= M + M
µ µc µd¶
Admissible measures always exist on a locally compact group. We expect
that the argumentspresented in thispaper would, with some modifications,
work for a larger class of measures. However, the above setting allows to
identifyacontinuousadmissiblemeasureµnaturally,viathedensityρ,with
anelementofthegroupalgebraC (G)(respectivelythegroupringCG,inthe
c
discretecase),whichiscrucialforfurtherapplications.
The set Q will usually be a Kazhdan set (see Definition 2.3). Since such
sets can be finite even for Lie groups (see Section 3.g), it is useful to work
with measures having an atomic part even in the Lie group setting. When
µ is continuous, hence entirely defined by a density ρ with respect to the
10 CORNELIADRUT¸UANDPIOTRW.NOWAK
Haar measure η, we sporadically replace µ by ρ in the whole notation and
terminology.
2.c. Groups and representations. Let G be a locally compact group. An
isometric representation π:G B(E) of G on a Banach space E is said to
→
becontinuousifitiscontinuouswithrespecttothestrongoperatortopology.
Equivalently,everyorbitmapiscontinuous,see[2,Lemma2.4]. Throughout
thearticlewerestrictourattentiontorepresentationsthatarecontinuousin
theabovesense,withoutmentioningthisfurther.
Considerthesubspaceof E consistingofvectorsinvariantunderπ,
Eπ v E : πgv vforevery g G .
= ∈ = ∈
The dual space E is ©naturally equipped with a contªragradient represen-
∗
tation π:G →I(E∗), defined by the formula πg =π∗g−1. Note that π is iso-
metric if π is, but not necessarily continuous. If E is reflexive we define a
subspaceE Ann (E )π ,whereAnndenotestheannihilator: theset ofall
π ∗
=
functionalsin E E that vanish identicallyon (E )π. Both Eπ and E are
= ¡∗∗ ¢ ∗ π
π–invariantclosedsubspacesof E.
Definition2.2. Arepresentationπ:G B(E)iscomplementedif
→
(5) E Eπ E .
π
= ⊕
Afamilyofcomplementedrepresentationsiscalledacomplementedfamily.
Examples of complemented representationsincludeisometric representa-
tions on reflexive Banach spaces [4] (in particular, on uniformly convex Ba-
nachspaces[2,Section2.c]),andrepresentationsofsmallexponentialgrowth
of certain Lie groups on Banach spaces with non-trivial Rademacher type
[35].
Arepresentationπ:G B(E)isuniformlyboundedif π supg G πg B(E)
. For any such represe→ntationπ, a new norm can be dkefikne=d on E∈, ekquikva- <
∞
lenttotheinitialone,bytheformula
(6) v π sup πgv .
k k = k k
g G
∈
Asobservedin[2,Proposition2.3],themodulusofconvexityofthenorm
π
k·k
satisfiesδ (t) δ t π 1 forevery t 0.
−
k·kπ ≥ k·k k k >
¡ ¢
2.d. Spectral gaps and uniform property (TE). Throughoutthesection,
G isalocallycompactgroupand E aBanachspace.
A representation π of G on E has almost invariantvectors if for every ε
>
0 and every compact subset S in G there exists v E, v 1, such that
∈ k k=
sups S v πsv ε.
∈ k − k≤
Definition2.3. (i) A complemented representationπ:G B(E) has a
→
spectralgap if therestrictionof π to E doesnot havealmost invari-
π
ant vectors, i.e. if there exists a constant κ 0 and a compact subset
>