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 projections 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 representations. 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 convergence, 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 property(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 conjecture. 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 isometric 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 representations 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 >

Kazhdan projections, random walks and ergodic theorems

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