aa r X i v : . [ m a t h . G R ] A p r PROPERTY (T) AND ACTIONS ON INFINITE MEASURESPACES
PAUL JOLISSAINT
Abstract.
The aim of the article is to provide a characterization of Kazhdan’sproperty (T) for locally compact, second countable pairs of groups H Ă G interms of actions on infinite, σ -finite measure spaces. It is inspired by therecent characterization of the Haagerup property by similar actions due to T.Delabie, A. Zumbrunnen and the author. Introduction
Throughout this article, G denotes a locally compact, second countable group(lcsc group for short); we assume furthermore that it is non-compact.In [6], the author and his co-authors T. Delabie and A. Zumbrunnen presenta characterization of the Haagerup property in terms of actions on infinite, σ -finite measure spaces having an invariant mean and whose associated permutationrepresentations are C .Citing A. Valette in Chapter 7 of [4], ”According to the philosophy that, to anycharacterization of property (T) there is a parallel characterization of the Haagerupproperty”, the aim of the present note is to propose a sort of reciprocal to thatstatement, namely, to propose a new characterization of property (T) inspired bythe above mentionned one of the Haagerup property.The objects under study here are what we call dynamical systems : given a lcscgroup G , such a dynamical system is a quadruple p Ω , B , µ, G q where p Ω , B , µ q is ameasure space on which G acts by µ -preserving automorphisms. For brevity, if µ is infinite, we denote by B f the subset of elements B P B such that 0 ď µ p B q ă 8 .Throughout the article, if p Ω , B , µ, G q is a dynamical system as above, we denoteby π Ω : G Ñ U p L p Ω , B , µ qq the associated permutation representation defined by p π Ω p g q ξ qp ω q : “ ξ p g ´ ω q for g P G and ω P Ω.Let p π, H q be a unitary representation of the lcsc group G . Then recall fromDefinition 1.1.1 of [2] that π almost has invariant vectors if, for every compact set H “ Q Ă G and for every ε ą
0, there exists a unit vector ξ P H such thatsup g P Q } π p g q ξ ´ ξ } ă ε. Recall also from Definition 1.1 of [9] that, if H is a closed subgroup of a lcscgroup G , then the pair H Ă G has Property (T) if, for every unitary representation Date : April 15, 2020.2010
Mathematics Subject Classification.
Primary 22D10, 22D40; Secondary 28D05.
Key words and phrases.
Locally compact groups, measure-preserving action, weakly mixingunitary representations, Property (T). π of G which almost has invariant vectors, there exists a vector ξ “ π p h q ξ “ ξ for every h P H . In particular, G has property (T) if the pair G Ă G hasproperty (T).Before stating our main result, we need two definitions. The first one is a weaken-ing of C - dynamical systems from [6]: recall from Definition 1.2 of [6] that a dynam-ical system p Ω , B , µ, G q is C if, for all A, B P B f , one has lim g Ñ8 µ p gA X B q “ Definition 1.1.
Let p Ω , B , µ, G q be a dynamical system as above. We say that itis weakly C if, for all A, B P B f and for any ε ą
0, there exists g P G such that µ p gA X B q ă ε .Next, the following type of sequence p B m q m ě Ă B f plays a crucial role in theproof of the main result of [6] (proof of Proposition 2.8): Definition 1.2.
Let p Ω , B , µ, G q be a dynamical system. A sequence p B m q m ě Ă B f is said to be almost invariant if µ p B m q “ m ě K Ă G , one haslim m Ñ8 sup g P K µ p gB m X B m q “ . Remark . Let p Ω , B , µ, G q be a weakly C dynamical system.(1) If A P B f is G -invariant then µ p A q “
0. In particular, a non-trivial weakly C dynamical system has an infinite measure.(2) As will be explained in more details in Section 2, the unitary representation π Ω is weakly mixing in the sense of [3], which means, by Theorem 1.9 of [3],that π Ω has no non-trivial finite-dimensional subrepresentation. Remark . Let p Ω , B , µ, G q be a dynamical system which contains an almost in-variant sequence of sets p B m q m ě Ă B f . Then the representation p π Ω , L p Ω , B , µ qq almost has invariant vectors namely, ξ m : “ χ B m P L p Ω , B , µ q is a unit vector forevery m ě Q Ă G and every ε ą
0, one hassup g P Q } π Ω p g q ξ m ´ ξ m } ă ε for every large enough m .Indeed, one has for all g and m } π Ω p g q ξ m ´ ξ m } “ p ´ Re x π Ω p g q ξ m | ξ m yq“ p ´ ż Ω χ gB m p ω q χ B m p ω q dµ p ω qq“ p ´ µ p gB m X B m qq which converges to 0 uniformly on Q as m Ñ 8 .Here is our main result.
Theorem 1.5.
Let G be a lcsc group and let H Ă G be a closed subgroup of G . (1) If the pair H Ă G has Property (T) and if p Ω , B , µ, G q is a dynamical sys-tem such that p Ω , B , µ, H q is weakly C , then p Ω , B , µ, G q admits no almostinvariant sequence. (2) If the pair H Ă G does not have Property (T), then there exists a σ -finite,dynamical system p Ω , B , µ, G q which has an almost invariant sequence andwhose restriction p Ω , B , µ, H q is weakly C . ROPERTY (T) AND INFINITE MEASURES 3
The proof of Theorem 1.5 will be given in Section 3, and it rests partly onweakly mixing representations and weakly mixing actions of groups on probabilitymeasure-preserving spaces, which is the subject of the next section.2.
Weakly mixing representations, weakly mixing actions
As mentioned in Section 1, we will see that the notion of weakly C dynami-cal systems is closely related to weakly mixing representations and weakly mixingactions.Our references for the latter properties are the article [3] of V. Bergelson andJ. Rosenblatt on the one hand, and Chapters 1 and 2 of the monograph [7] by E.Glasner on the other hand. Notation
Let G be a lcsc group and let p S, B S , ν q be a probability space on which G acts by Borel automorphisms and preserves ν . Then the subset L p S, ν q : “ " ξ P L p S, ν q : ż S ξdν “ * is a G -invariant closed subspace of L p S, ν q , and we denote by π S, the restrictionof π S to L p S, ν q . Definition 2.1.
Let G be a lcsc group.(1) A unitary representation p π, H q of G is weakly mixing if it contains nonon-trivial finite-dimensional subrepresentation.(2) Let p S, B S , ν, G q be a dynamical system where ν is a G -invariant proba-bility measure. We say that the action of G on S is weakly mixing if therepresentation π S, is a weakly mixing representation. Remark . The original definition of weakly mixing representations involve thespace of weakly almost periodic functions
W AP p G q on G and the unique invariantmean on it (cf. [3], Definition 1.1, and [7], Definition 3.2), and the characterizationin terms of finite-dimensional subrepresentations is for instance Theorem 1.9 of [3].We have chosen not to introduce W AP p G q because we feel that it is useless in thepresent context.We gather some characterisations of weakly mixing representations that will beused in the next section. Proposition 2.3.
Let G be a lcsc group, let p π, H q be a unitary representation of G and let T be a total subset of H . The following conditions are equivalent: (a) p π, H q is a weakly mixing representation of G ; (b) for every ε ą and for all ξ , . . . , ξ m P H , there exists g P G such that |x π p g q ξ j | ξ j y| ă ε for all j “ , . . . , m ; (c) for every ε ą and for all ξ , . . . , ξ m P T , there exists g P G such that |x π p g q ξ j | ξ k y| ă ε for all j, k “ , . . . , m .Proof. Equivalence between (a) and (b) follows from Corollary 1.6 and from The-orem 1.9 of [3], and equivalence between (b) and (c) is a consequence of densityof the span of T and of polar decomposition of scalar products in Hilbert spaces; PAUL JOLISSAINT namely, x ξ | η y “ px ξ ` η | ξ ` η y ´ x ξ ´ η | ξ ´ η y` i x ξ ` iη | ξ ` iη y ´ i x ξ ´ iη | ξ ´ iη yq . for all ξ, η P H . (cid:3) Corollary 2.4.
Let G be a lcsc group and let p Ω , B , µ, G q be a dynamical system.Then the following conditions are equivalent: (a) p Ω , B , µ, G q is weakly C ; (b) the permutation representation π Ω is weakly mixing.Proof. Assume that condition (a) holds. This means that for all
A, B P B f , and forevery ε ą
0, there exists g P G such that |x π Ω p g q χ A | χ B y| ă ε . If A , . . . , A m P B f and if ε ą
0, then considering A “ B “ Ť mj “ A j which belongs to B f , there exists g P G such that |x π Ω p g q χ A j | χ A k y| ă ε for all j, k “ , . . . , m . As linear combinationsof characteristic functions χ A with A P B f are dense in L p Ω , B , µ q , this proves thatcondition (b) holds.Conversely, if (b) holds, then (a) is the special case m “ ξ “ χ A and ξ “ χ B in condition (c) of Proposition 2.3. (cid:3) Lemma 2.5.
Let p S, B S , ν q be a probability space on which G acts by ν -preservingBorel automorphisms, and assume that the action is weakly mixing. Then, for every ε ą and for all A , . . . , A m P B S , there exists g P G such that | ν p gA j X A k q ´ ν p A j q ν p A k q| ă ε for all j, k “ , . . . , m .Proof. For j “ , . . . , m , set ξ j : “ χ A j ´ ν p A j q . Then ξ j P L p S, ν q for every j , and,for every ε ą
0, by Proposition 2.3, there exists g P G such that |x π S, p g q ξ j | ξ k y| ă ε for all j, k “ , . . . , m . But we have x π S, p g q ξ j | ξ k y “ x χ gA j ´ ν p A j q| χ A k ´ ν p A k qy“ ν p gA j X A k q ´ ν p A j q ν p A k q ` ν p A j q ν p A k q“ ν p gA j X A k q ´ ν p A j q ν p A k q for all j, k “ , . . . , m . (cid:3) Proof of Theorem 1.5
Let H Ă G be a pair of lcsc groups with Property (T) as in statement (1) of Theo-rem 1.5, and let p Ω , B , µ, G q be a dynamical system whose restriction to p Ω , B , µ, H q is weakly C . Corollary 2.4 implies that the permutation representation π Ω of H is weakly mixing, hence that it does not have any non-trivial finite-dimensionalsubrepresentation. If there existed an almost invariant sequence of sets p B m q Ă B f as in Definition 1.2, then the unitary representation π Ω of G would almost haveinvariant vectors, hence there would exist a non-zero vector ξ such that π Ω p h q ξ “ ξ for every h P H by Property (T), but this contradicts the weakly mixing propertyof π Ω restricted to H . Hence p Ω , B , µ, G q has no almost invariant sequence.The rest of the present section is devoted to the proof of statement (2) of Theorem1.5. Thus we assume henceforth that the pair H Ă G does not have Property (T).Let us choose an increasing sequence of compact subsets p K n q n ě of G with the ROPERTY (T) AND INFINITE MEASURES 5 following properties: e P ˚ K , K n Ă ˚ K n ` for every n ě G “ Ť n ě K n .Hence, K is a compact neighbourhood of e , and for every compact set K Ă G ,there exists m such that K Ă K m . Lemma 3.1.
With the above hypotheses, there exists a conditionally negative def-inite function ψ : G Ñ R ` whose restriction to H is unbounded.Proof. By Theorem 1.2 of [9], there exists a sequence p ψ n q n ě of real-valued, pos-itive definite and normalized functions on G which converges to 1 uniformly oncompacts sets, but such that sup h P H | ψ n p h q ´ | Ñ . Extracting subsequences if necessary, we assume that there exists c ą p h n q n ě Ă H such that 1 ´ ψ n p h n q ě c andsup g P K n | ψ n p g q ´ | ď n for all n . Then set ψ “ ÿ n ě ? n p ´ ψ n q . It defines a conditionally negative definite function on G which satisfies ψ p h k q “ ÿ n “ k ? n p ´ ψ n p h k qq ` ? k p ´ ψ k p h k qq ě ? kc. Hence ψ is unbounded on H . (cid:3) An adaptation of Proposition 2.2.3 of [4], of [5] and of Theorem A.1 of [8] showsthat there exists a measure-preserving G -action on a standard probability space p S, B S , ν q with the following properties:(a) the restriction to H of the action is weakly mixing (this is where the exis-tence of ψ in Lemma 3.1 is needed; see details below);(b) there exists a non-trivial asymptotically invariant sequence of Borel subsetsof S , namely, there exists a sequence p A n q n ě Ă B S such that ν p A n q “ { n and such that, for every compact set K Ă G ,lim n Ñ8 sup g P K ν p gA n △ A n q “ A △ B “ A z B Y B z A for all sets A, B .Furthermore, the proof of Lemma 1.3 of [1] shows that we can (and will) assumethat S is a compact metric space on which G acts continuously, and that ν hassupport S .We think that it is helpful to describe the construction of the probability G -space p S, B S , ν q with some details. We follow faithfully the proof of Theorem 2.2.2 of [4].By Lemma 3.1, let ψ : G Ñ R ` be a conditionally negative definite functionwhich is unbounded on H . For n ě
1, we set ϕ n “ exp p´ ψ { n q , and we denote by p π n , H n , ξ n q the associated Gel’fand-Naimark-Segal triple. Since ϕ n is real-valued,there exists a real Hilbert subspace H n of H n , containing ξ n , such that H n “ H n ‘ i H n and π n p g q H n “ H n PAUL JOLISSAINT for all n and g . We set H “ À n ě H n , H “ À n ě H n so that H “ H ‘ i H , and π “ À n ě π n . Finally, we identify ξ n with the corresponding vector0 ‘ . . . ‘ ξ n ‘ . . . P H and we observe that ξ n K ξ m when n “ m .Next, set H σ “ À k ě H ˆ k , where H ˆ “ C , and for k ą H ˆ k is the k -thsymmetric tensor product of H , that is, the closed subspace of the Hilbert tensorproduct space H b k generated by the vectors of the form ÿ s P S k η s p q b . . . b η s p k q where S k denotes the usual permutation group. Then the representation π extendsin a natural way to a representation π σ of G on H σ which leaves the subspace H σ “ H σ a H ˆ invariant. Finally, we denote by π σ the restriction of π σ to H σ . Lemma 3.2.
The restriction of π σ to H is weakly mixing.Proof. Let p h ℓ q ℓ ě Ă H be a sequence such that ψ p h ℓ q Ñ 8 as ℓ Ñ 8 . Then, asin the proof of Lemma 2.1 of [9], for every finite set F Ă G , one hasmax g,g P F ψ p gh ℓ g q Ñ 8 as ℓ Ñ 8 . As the set of vectors t π n p g q ξ n : g P G, n ě u is total in H , it suffices, byProposition 2.3, to prove that, for all g , . . . , g m P G , for every n ě ε ą ℓ ě |x π n p g j h ℓ g k q ξ n | ξ n y| ă ε for all j, k . But x π n p g j h ℓ g k q ξ n | ξ n y “ exp p´ ψ p g j h ℓ g k q{ n q Ñ ℓ Ñ 8 . (cid:3) In order to define p S, ν q , we choose a countable orthonormal basis B of H whichcontains t ξ n : n ě u , we set p S, ν q : “ ź b P B ´ R , ? π exp ´ ´ x ¯ dx ¯ and we define the random variable X b : S Ñ R by X b pp ω b q b P B q “ ω b for every b .Then the map ξ “ ÿ b P B ξ b b ÞÑ ÿ b P B ξ b X b from H to L p S, ν q extends to a surjective isometry u : H σ Ñ L p S, ν q which sends H ˆ onto the space of constant functions on S and such that u ´ ÿ s P S n b s p q b . . . b b s p n q ¯ “ n ! X b . . . X b n for all b , . . . , b n P B . Moreover, there is a ν -preserving action of G on p S, ν q suchthat u ˚ π S p g q u “ π σ p g q and u ˚ π S, p g q u “ π σ p g q for all g P G . In particular, Lemma3.2 implies that the action of H on p S, ν q is weakly mixing. Finally, the non-trivialasymptotically invariant sequence p A n q Ă B S of condition (b) above is obtained bysetting A n “ t ω P S : X ξ n p ω q ě u . See pages 23 and 24 of [4] for further details.Let us describe now our construction of the dynamical system p Ω , B , µ, G q whichis taken from [6]. ROPERTY (T) AND INFINITE MEASURES 7
Let p A n q n ě Ă B S be the above non-trivial asymptotically invariant sequence.Then the following inequalities | ν p gA n X A n q ´ { | “ | ν p gA n X A n q ´ ν p A n q|“ ż S χ A n p χ gA n ´ χ A n q dν ď ż S | χ gA n ´ χ A n | dν “ ν p gA n △ A n q show that, for all positive integers m and k , there exists an integer n p k, m q suchthat sup g P K m ˇˇˇˇ ν p gA n p k,m q X A n p k,m q q ´ ˇˇˇˇ ď ´ ´ e ´ m k ¯ . Then set B m : “ ź k ě A n p k,m q Ă X : “ ź k ě S for every m ą X with a σ -algebra C containing p B m q m ě and with a measure µ : C Ñ r , which have the following properties (see [6], Propositions 2.8, 3.4and 3.5):(i) The σ -algebra C is generated by the collection (denoted by F c, in Definition3.1 of [6]) of sets C “ ś n C n such that ś n ν p C n q : “ lim N Ñ8 ś Nn “ ν p C n q exists in r , and such that ś n ν p C n q “ N Ñ8 8 ź n “ N ν p gC n X C n q “ g P K .(ii) The measure µ on C satisfies µ p C q “ ź n ν p C n q for every set C “ ś n C n P F c, .(iii) The diagonal action of G on X is C -measurable.(iv) For every A P C such that µ p A q ă 8 , one haslim g Ñ e µ p gA △ A q “ . Remark . (1) The measure µ on X is not necessarily σ -finite, as is provedin Proposition 2.6 of [6]. Thus, equality (3.1) in (i) and property (iv) arecontinuity properties that are needed to restrict µ to a G -invariant subsetΩ P C on which is it σ -finite: see Section 3 of [6].(2) We observe that the sequence p B m q m ě is contained in C : indeed, as e ´ m k ď ν p gA n p k,m q X A n p k,m q q ď PAUL JOLISSAINT for every g P K m and for every k ě
1, we get that1 ě lim N Ñ8 8 ź k “ N ν p gA n p k,m q X A n p k,m q qě lim N Ñ8 8 ź k “ N e ´ m k “ e ´ m lim N Ñ8 ř k “ N k “ . In particular, it converges uniformly for g P K .The following proposition is inspired by Proposition 2.8 of [6]. Proposition 3.4.
The dynamical system p X, C , µ, H q is weakly C , namely, for all A, B P C such that µ p A q , µ p B q ă 8 and for every ε ą , there exists h P H suchthat µ p hA X B q ă ε .Proof. Assume first that
A, B P F c, have positive measures, and write A “ ś n A n and B “ ś n B n , so that µ p A q “ ź n ě ν p A n q and µ p B q “ ź n ě ν p B n q . Let ε ą ε ą δ : “ ` ε ` ε ´ ε ă . Since 0 ă µ p A q , µ p B q ă 8 , there exists N large enough such that12 ´ ε ă ν p A n q , ν p B n q ă ` ε @ n ě N. Since δ ă
1, there exists m large enough such that δ m ` ă ε { µ p A q . The action of H on p S, ν q being weakly mixing, by Lemma 2.5, there exists h P H such that | ν p hA n X B n q ´ ν p A n q ν p B n q| ď ε for all n P t
N, . . . , N ` m u . Then we have µ p hA X B q ď N ´ ź n “ ν p A n q ¨ N ` m ź n “ N p ν p A n q ν p B n q ` ε q ¨ ź n ě N ` m ` ν p A n q“ µ p A q ¨ N ` m ź n “ N ν p A n q ν p B n q ` ε ν p A n q“ µ p A q N ` m ź n “ N ˆ ν p B n q ` ε ν p A n q ˙ ă µ p A q N ` m ź n “ N ˆ ` ε ` ε ´ ε ˙ “ µ p A q δ m ` ă ε. Hence the claim holds for all sets
A, B P F c, .The same claim holds for A, B which belong to the semiring F c generated by F c, because, by Proposition 3.4 of [6], for all A, B P F c , there exist C, D P F c, such that A Ă C and B Ă D . ROPERTY (T) AND INFINITE MEASURES 9
Moreover, it also holds for all elements of the ring R p F c q generated by F c .Finally, if A, B P C are such that 0 ă µ p A q , µ p B q ă 8 , if ε ą p C k q k ě , p D ℓ q ℓ ě Ă R p F c q such that A Ă ď k ě C k and B Ă ď ℓ ě D ℓ and µ p A q ď ÿ k µ p C k q ă µ p A q ` ε and µ p B q ď ÿ ℓ µ p D ℓ q ă µ p B q ` ε. Choose first N large enough so that ř ℓ ą N µ p D ℓ q ă ε {
3. Then, as gA X B Ă ˜ N ď ℓ “ gA X D ℓ ¸ Y ˜ ď ℓ ą N D ℓ ¸ , we get µ p gA X B q ď N ÿ ℓ “ µ p gA X D ℓ q ` ε { g P G .Choose next M large enough so that ř k ą M µ p C k q ă ε { N . Then, as gA X D ℓ Ă ˜ M ď k “ gC k X D ℓ ¸ Y ˜ ď k ą M gC k ¸ for every 1 ď ℓ ď N , and since µ is G -invariant, we get µ p gA X B q ď N ÿ ℓ “ M ÿ k “ µ p gC k X D ℓ q ` ε { g P G . By the previous part of the proof applied to C : “ Ť Mk “ C k and D : “ Ť Nℓ “ D ℓ , there exists h P H such that µ p hA X B q ă ε. (cid:3) The proof of Theorem 1.5 will be complete if we prove that there is a G -invariantset Ω P C on which µ is σ -finite.As in Proposition 3.8 of [6], we fix a countable dense subset D Q e of G , we set Y “ Ť m ě B m and Ω “ ď h P D hY on which µ is obviously σ -finite.The continuity condition (iv) implies that Ω is G -invariant (see the proof ofProposition 3.8 of [6]), thus the proof of Theorem 1.5 is complete. References [1] S. Adams, G. A. Elliot, and T. Giordano. Amenable actions of groups.
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Universit´e de Neuchˆatel, Institut de Math´ematiques, E.-Argand 11, 2000 Neuchˆatel,Switzerland
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