On complements of Kazhdan projections in semisimple groups
aa r X i v : . [ m a t h . G R ] J un On complements of Kazhdan projections insemisimple groups *Piotr W. Nowak † and Eric Reckwerdt ‡ Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
June 22, 2018
Abstract
We prove that for an isometric representation of some groups on cer-tain Banach spaces, the complement of the subspace of invariant vectorsis 1-complemented.
An isometric representation π of a locally compact group G on a reflexiveBanach space E induces a direct sum decomposition E = E π ⊕ E π into the invariant vectors E π and its canonical, π -invariant complement, E π .The corresponding projection P π from E to E π along E π has norm 1 underfairly general conditions. However, typically, the complementary projection I − P π need not be of norm 1. An easy example is that of the regular rep-resentation of a finite group G on ℓ p ( G ), p >
2, where E π are the constantfunctions and E π are the functions on G whose mean is zero (see Example 4for details).In this article we give conditions under which, for representations π on aBanach space E in a certain class O , consisting of uniformly convex uniformly * This project has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (grant agreement no.677120-INDEX). † [email protected] ‡ [email protected] E π is 1-complemented and more precisely that k I − P π k = Theorem 1.
Let G be a locally compact group and let π be an isometric rep-resentation on a Banach space E ∈ O . Assume that there exists a sequence ofelements { g n } ⊂ G, g n → ∞ , such that WOT − lim π g n = WOT − lim π g − n = P π . Then E π is Birkhoff-James orthogonal to E π and k I − P π k = . The above theorem in particular applies to class of groups satisfying cer-tain algebraic conditions, introduced as quasi-semisimple groups in [2]. Theyare defined via the existence of a
K AK -type decomposition and generationby certain contraction subgroups, see [2, 9] for details. This class of groupsincludes the classical semisimple Lie groups as well as certain automorphismgroups of trees.Recall that a representation π of G on a reflexive Banach space E is a c -representation if the matrix coefficients 〈 π g v , w 〉 vanish at infinity for every v ∈ E π and w ∈ E ∗ . As shown in [2], for quasi-semisimple groups this is thecase for all isometric representations on reflexive Banach spaces. Corollary 2.
Let π be a representation of a non-compact locally compactgroup G on E ∈ O . If π is a c -representation then k − P π k = .In particular, if G is quasi-semisimple then the above estimate holds forall isometric representations on E ∈ O . This is particularly interesting in the context of properties such as prop-erty ( T ℓ p ), studied by Bekka and Olivier [4]. In that case we obtain a strongerconclusion that holds not only in the algebra of bounded operators on E ∈ O ,but in the the associated maximal group Banach algebra (see Section 3.1). Corollary 3.
Let G be quasi-semisimple. Assume that G satisfies uniformproperty ( T O ) , namely that there exists a Kazhdan projection p in the Banachalgebra C O max ( G ) . Then the element − p satisfies k − p k = . We remark that in [5] Bernau and Lacey studied projections P such thatboth P and I − P are of norm 1, and in particular gave a general descriptionof such projections in L p -spaces. 2 cknowledgements We are very grateful to Nicolas Monod and Mikael de la Salle for insightfulcomments on an earlier version of the preprint and to Uri Bader for illumi-nating comments and discussions.
Consider a duality mapping J : E → S ( E ∗ ), into the collection S ( E ∗ ) of sub-sets of E ∗ , defined by the condition J ( v ) = © w ∈ E ∗ : 〈 v , w 〉 = k v k and k w k = k v k ª .A thorough discussion of duality mappings in Banach spaces can be found ine.g. [8].Denote by w − lim the limit in the weak topology on E . A Banach space E has the Opial property if for every weakly convergent sequence { x n } ⊆ E withthe weak limit x ∈ E the inequalitylim inf k x n − x k < lim inf k x n − y k ,holds for every y x . Every separable Banach space admits an equivalentnorm with the Opial property [11].It is known that a uniformly convex uniformly smooth Banach space hasOpial’s property if and only if the following condition is satisfied: for everysequence { x n } in E , we have w − lim x n = x ⇐⇒ w − lim J ( x n − x ) = ∆ -convergence. We refer to[18] for details.By O we will denote the class of uniformly convex uniformly smooth Ba-nach spaces with the Opial property. It is known for instance that the spaces ℓ p , 1 < p < ∞ , belong to the class O , as do their infinite direct q -sums,1 < q < ∞ . However, the spaces L p [0, 1], p ∈ (1, 2) ∪ (2, ∞ ) do not belong tothe class O . 3 .2 Representations Let B ( E ) denote the space of bounded linear operators on E . We will consider B ( E ) with the strong operator topology ( SOT ) and with the weak operatortopology (
WOT ). Let G be a locally compact group. Let π be an isometricrepresentation of G on E ∈ O that is continuous in the strong operator topol-ogy on B ( E ). In other words, for every v ∈ E the orbit map G → E , g π g v ,is continuous into the norm topology on E . The dual space E ∗ is naturallyequipped with an isometric representation π , defined by the formula π g = ( π ∗ g ) − ,for every g ∈ G . If π is SOT -continuous then so is π , under the assumptionthat E is reflexive, see [20, Proposition 4.1.2.3, page 224]. A matrix coefficient of a representation π on a Banach space E , associated to vectors v ∈ E , w ∈ E ∗ ,is a function ψ v , w : G → C defined by ψ v , w ( g ) = 〈 π g v , w 〉 . When E is reflexivewe will say that π is a c -representation if ψ v , w ∈ C ( G ) for every v ∈ E π and w ∈ E ∗ . Denote E π = © v ∈ E : π g v = v for all g ∈ G ª .If E is reflexive then there is a direct sum decomposition E = E π ⊕ E π , where E π is the annihilator of the space ( E ∗ ) π . Since E π is π -invariant, the decom-position is in fact a decomposition of π into the trivial representation and itscomplement. See e.g. [1, 3]. Note however, that if E is not reflexive then theprojection onto E π might not be equivariant [1, Example 2.29], and in factthere might not be a bounded projection onto E π at all [17, Theorem 1].Let P π : E → E π be the projection along E π . The projection P π is known tosatisfy k P π k = k I − P π k ≤ Example 4.
Let G be a finite group. Consider the left regular representationof G on E = ℓ p ( G ), 1 ≤ p ≤ ∞ . The invariant vectors are then the constantfunctions on G and the complement E π is the subspace of functions satisfying M f = G X h ∈ G f ( h ) = p = ∞ and the group Z n , n ≥ k I − P π k > f = (1, 1, . . ., 1, − ∈ ℓ ∞ ( Z n ). Then k f k = M f = − n and k f − M f k = − n . Since G is finite, clearly, by choosing n ∈ N and 2 < p <∞ , sufficiently large we obtain that the norm of k I − P π k can be arbitrarilyclose to 2. The same is true for any group with a finite quotient of cardinalityat least 3.Observe that if G is residually finite with a sequence { N i } of finite indexnormal subgroups, T N i = { e } , then the previous case shows that for the rep-resentation of G on ℓ p ( ` G / N i ) = ( L ℓ p ( G / N i )) ( p ) for p = ∞ the norm k I − P π k will be in fact 2. We can thus choose p such that the norm of the projection I − P π will be arbitrarily close to 2. Before proving the main theorem we first need a few lemmas regarding du-ality mappings and their behavior with respect to isometric representationswithin the class O of Banach spaces. For an invertible isometry S on E denote S = ( S ∗ ) − . Lemma 5.
Let E ∈ O be a Banach space and let S ∈ B ( E ) be an invertibleisometry. Then S J ( v ) = J ( Sv ) .Proof. First note that since S is an invertible isometry, so is S ∗ , and we havethat k S ∗ J ( Sv ) k = k J ( Sv ) k = k Sv k = k v k . Thus 〈 v , S ∗ J ( Sv ) 〉 = 〈 Sv , J ( Sv ) 〉 = k Sv k = k v k = 〈 v , J ( v ) 〉 .By uniform convexity and uniform smoothness of E the duality mapping J isbijective, which guarantees that there is a unique element w ∈ E ∗ such that k w k = k v k and 〈 v , w 〉 = k v k . This yields S ∗ J ( Sv ) = J ( v ),and the claim is proved.A vector 0 v ∈ E is Birkhoff-James orthogonal to 0 w ∈ E if k v k ≤ k v + λ w k for every λ ∈ R . Given subspaces V , V in E we will say that V is Birkhoff-James orthogonal to V if for every v ∈ V and v ∈ V , v is Birkhoff-Jamesorthogonal to v . 5 useful lemma by Kato [13] states that k v k ≤ k v + λ w k for every λ > 〈 w , J ( v ) 〉 ≥
0. (1)In particular this implies that v is Birkhoff-James orthogonal to w if we have 〈 w , J ( v ) 〉 ≥ 〈 w , J ( − v ) 〉 ≥
0. Of course, this is equivalent to 〈 w , J ( v ) 〉 = Theorem 1.
Let G be a locally compact group and let π be an isometric rep-resentation on a Banach space E ∈ O . Assume that there exists a sequence ofelements { g n } ⊂ G, g n → ∞ , such that WOT − lim π g n = WOT − lim π − g n = P π . Then E π is Birkhoff-James orthogonal to E π and k I − P π k = .Proof. Let w ∈ E π , v ∈ E π be arbitrary. We first observe that since P π = WOT − lim π g n , then ( P π ) ∗ = WOT − lim π ∗ g n = WOT − lim π − g n .Thus we have 〈 w , J ( v ) 〉 = 〈 P π w , J ( v ) 〉= 〈 w , ( P π ) ∗ J ( v ) 〉= lim D w , π g − n J ( v ) E = lim D w , J ( π g − n v ) E ,where the last equality follows from Lemma 5.Now, by assumption, the sequence π − g n v converges weakly to P π v = v ∈ E π . By the Opial property we have that this is the same as the conditionthat J ( π − g n v ) → E ∗ . In particular,lim w , J ( π g n v ) ® = 〈 w , J ( v ) 〉 = k I − P π k = quasi-semisimple groups , stud-ied in [2]. These groups are defined as satisfying two algebraic conditions:a version of a K AK -decomposition and that the group is generated by cer-tain contraction subgroups, associated to sequences of elements tending offto infinity. We refer to [2] and [9] for definitions and applications.6 roof of Corollary 2. If π is an isometric c -representation of G on a reflexiveBanach space E then, for v ∈ E π and w ∈ E ∗ , every matrix coefficient of π of the form ψ v , w = 〈 π g v , w 〉 vanishes as g goes to infinity. This in particularmeans, that the WOT closure of π ( G ) in B ( E ) is the one-point compactificationof G , with the point at infinity being precisely the projection P π . In particular,this ensures that the assumptions of Theorem 1 are satisfied.The class of quasi-semisimple groups includes the classical semisimpleLie groups and certain automorphism groups of trees. As proved in [2],quasi-semisimple groups satisfy a Veech decomposition of their space W ( G )of weakly almost periodic functions, W ( G ) = C ( G ) ⊕ C .(See also [19] for the classical result of Veech on semisimple Lie groups.) Ev-ery matrix coefficient of an isometric representation on a reflexive Banachspace E is an element of W ( G ), with the projection (invariant mean on weaklyalmost periodic functions) m : W ( G ) → C given by m ( ψ v , w ) = 〈 P π v , P π w 〉 . Thus,the previous argument applies to every isometric representation π of a quasi-semisimple group on E ∈ O . Let O be a class of Banach spaces. The group ring C G can be completed inthe the norm k f k = sup k π ( f ) k ,where the supremum is taken over all isometric representations of G on Ba-nach spaces E ∈ O . This completion is a Banach algebra, which we denote C O max ( G ). In the case when O consists of the class of Hilbert spaces the alge-bra is the maximal group C ∗ -algebra of G . A Kazhdan projection p ∈ C O max ( G )is an idempotent such that π ( p ) = P π for every isometric representation of G on E ∈ O . G has uniform property ( T O ) if a Kazhdan projection exists in p ∈ C O max ( G ). See [10] for a detailed study. Proof of Corollary 3.
Follows from the fact that for every isometric represen-tation of G on E ∈ O we have k I − P π k = References [1] U. Bader, A. Furman, T. Gelander, and N. Monod,
Property (T) and rigidity for actions onBanach spaces , Acta Math. (2007), no. 1, 57–105.
2] U. Bader and T. Gelander,
Equicontinuous actions of semisimple groups , Groups Geom.Dyn. (2017), no. 3, 1003–1039.[3] U. Bader, C. Rosendal, and R. Sauer, On the cohomology of weakly almost periodic grouprepresentations , J. Topol. Anal. (2014), no. 2, 153–165.[4] B. Bekka and B. Olivier, On groups with property ( T ℓ p ), J. Funct. Anal. (2014), no. 3,643–659.[5] S. J. Bernau and H. Elton Lacey, Bicontractive projections and reordering of L p -spaces ,Pacific J. Math. (1977), no. 2, 291–302.[6] F. E. Browder, Fixed point theorems for nonlinear semicontractive mappings in Banachspaces , Arch. Rational Mech. Anal. (1966), 259–269.[7] N. Brown and E. Guentner, Uniform embeddings of bounded geometry spaces into reflex-ive Banach space , Proc. Amer. Math. Soc. (2005), no. 7, 2045–2050.[8] C. Chidume,
Geometric properties of Banach spaces and nonlinear iterations , LectureNotes in Mathematics, vol. 1965, Springer-Verlag London, Ltd., London, 2009.[9] C. Ciobotaru,
A unified proof of the Howe-Moore property , J. Lie Theory (2015), no. 1,65–89.[10] C. Dru¸tu and P. W. Nowak, Kazhdan projections, random walks and ergodic theorems ,Crelle’s Journal, to appear.[11] D. van Dulst,
Equivalent norms and the fixed point property for nonexpansive mappings ,J. London Math. Soc. (2) (1982), no. 1, 139–144.[12] F. J. García-Pacheco, Complementation of the subspace of G-invariant vectors , J. AlgebraAppl. (2017), no. 7, 1750124, 7.[13] T. Kato, Nonlinear semigroups and evolution equations , J. Math. Soc. Japan (1967),508–520.[14] J. Lamperti, On the isometries of certain function-spaces , Pacific J. Math. (1958), 459–466.[15] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I , Springer-Verlag, Berlin-NewYork, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol.92.[16] Zdzisław Opial,
Weak convergence of the sequence of successive approximations for nonex-pansive mappings , Bull. Amer. Math. Soc. (1967), 591–597. MR0211301[17] T. Shulman, On subspaces of invariant vectors , Studia Math. (2017), no. 1, 1–11.[18] S. Solimini and C. Tintarev,
Concentration analysis in Banach spaces , Commun. Con-temp. Math. (2016), no. 3, 1550038, 33.[19] W. A. Veech, Weakly almost periodic functions on semisimple Lie groups , Monatsh. Math. (1979), no. 1, 55–68.[20] G. Warner, Harmonic analysis on semi-simple Lie groups. I , Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188., Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188.