Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions
aa r X i v : . [ m a t h . D S ] F e b HAAGERUP PROPERTY AND KAZHDAN PAIRS VIAERGODIC INFINITE MEASURE PRESERVING ACTIONS
Alexandre I. Danilenko
Abstract.
It is shown that a locally compact second countable group G has theHaagerup property if and only if there exists a sharply weak mixing 0-type mea-sure preserving free G -action T = ( T g ) g ∈ G on an infinite σ -finite standard mea-sure space ( X, µ ) admitting a T -Følner sequence (i.e. a sequence ( A n ) ∞ n =1 of mea-sured subsets of finite measure such that A ⊂ A ⊂ · · · , S ∞ n =1 A n = X andlim n →∞ sup g ∈ K µ ( T g A n △ A n ) µ ( A n ) = 0 for each compact K ⊂ G ). A pair of groups H ⊂ G has property (T) if and only if there is a µ -preserving G -action S on X admitting an S -Følner sequence and such that S ↾ H is weakly mixing. These refinesome recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint.
0. Introduction
Throughout this paper G is a non-compact locally compact second countablegroup. It has the Haagerup property if there is a weakly continuous unitary repre-sentation V of G in a separable Hilbert space H such that lim g →∞ V ( g ) = 0 in theweak operator topology and( ∗ ) for each ǫ > K ⊂ G , there is a unit vector ξ ∈ H such that sup g ∈ G k V ( g ) ξ − ξ k < ǫ .Of course, the amenable groups have the Haagerup property. The class of discretecountable Haagerup groups contains the free groups and is closed under free prod-ucts and wreath products [CoStVa]. For more information about the Haagerupproperty we refer to [Ch–Va]. There is a purely dynamical description of thisproperty: G is Haagerup if and only if there exists a mixing non-strongly ergodicprobability preserving free G -action [Ch–Va, Theorem 2.2.2] (see § Theorem A. G has the Haagerup property if and only if there is a 0-type measurepreserving G -action T = ( T g ) g ∈ G on an infinite σ -finite measure space ( X, B , µ ) admitting a sequence of non-negative unit vectors ( ξ n ) ∞ n =1 in L ( X, µ ) such that lim n →∞ sup g ∈ K h ξ n ◦ T g , ξ n i = 1 for each compact K ⊂ G . We recall that T is called of 0-type if lim g →∞ µ ( T g A ∩ B ) = 0 for all subsets A, B ∈ B of finite measure. In this paper we provide a much shorter alternativeproof of Theorem A which is grounded on the Moore-Hill concept of restrictedinfinite products of probability measures [Hi].We note that the 0-type for infinite measure preserving systems is a natural coun-terpart of the mixing for probability preserving systems. However unlike mixing, Typeset by
AMS -TEX Theorem B.
The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preservingfree G -action T on an infinite σ -finite standard measure space admitting anexhausting T -Følner sequence of subsets. (iii) There exists a a sharply weak mixing (conservative) 0-type measure preserv-ing free G -action T on an infinite σ -finite standard measure space ( X, B , µ ) admitting a T -Følner sequence ( A n ) ∞ n =1 such that µ ( A n ) = 1 for all n ∈ N . We say that ( A n ) ∞ n =1 is T -Følner if µ ( A n ) < ∞ and sup g ∈ K µ ( A n △ T g A n ) µ ( A n ) → n → ∞ for each compact subset K ⊂ G . If A ⊂ A ⊂ · · · and S ∞ n =1 A n = X ,we say that ( A n ) ∞ n =1 is exhausting . We note that sharp mixing (see § G -action from [Ch–Va, Theorem 2.2.2] (cf. the construction of II ∞ ergodicPoisson suspensions of countable amenable groups from [DaKo]). Then we observethat the action T that we obtain is IDPFT (see § T is sharply weakmixing whenever we show that it is conservative. To show the conservativenessof T is remains to choose the parameters of the Moore-Hill construction in an aappropriate way.As a corollary from Theorem B, we obtain one more dynamical characterizationof the Haagerup property in terms of Poisson actions. Corollary C. G has the Haagerup property if and only if there exists a mixing(probability preserving) Poisson G -action that is not strongly ergodic. Our next purpose is to obtain a “parallel” characterization of property (T) whichis a reciprocal to the Haagerup property. We recall [Jo1, Definition 1.1] that givena non-compact closed subgroup H of G , the pair H ⊂ G has property (T) if for eachunitary representation V of G satisfying ( ∗ ), there is a unit vector which is invariantunder V ( h ) for every h ∈ H . Using the techniques developed for proving Theorem Bwe obtain an ergodic (non-spectral) characterization of Kazhdan pairs that refinesa spectral characterization from [Jo2]. Theorem D. (i)
If a pair H ⊂ G has property (T) then each measure preserving G -action S = ( S g ) g ∈ G on a σ -finite infinite standard measure space ( Y, C , ν ) , suchthat S ↾ H := ( S h ) h ∈ H has no invariant subsets of positive finite measure,admits no S -Følner sequences. Conservativeness, ergodicity, weak mixing and sharp weak mixing are not spectral invariants ofthe underlying dynamical systems. Hence the principal difference of Theorem B from Theorem Ais that it provides non-spectral ergodic characterization of the Haagerup property.
If a pair H ⊂ G does not have property (T) then there is a measure preserv-ing G -action S on a σ -finite infinite measure space which has an exhausting S -Følner sequence and such that S ↾ H is weakly mixing. Let us say that S ↾ H is of weak 0-type if there is a subsequence h n → ∞ in H such that lim n →∞ ν ( S h n A ∩ B ) = 0 for all subsets A, B ∈ C of finite measure.Then replacing the “has no invariant subsets of positive finite measure” in (i) witha stronger “is of weak 0-type”, and the “weakly mixing” in (ii) with a weaker “ofweak 0-type” we obtain exactly [Jo2, Theorem 1.5]. Corollary E.
A pair H ⊂ G has property (T) if and only if every (probabilitypreserving) Poisson G -action with weakly mixing H -subaction is strongly ergodic.The same is also true with “ergodic” in place of “weakly mixing”. The outline of the paper is as follows. In Section 1 we state all necessary defini-tions related to the basic dynamical concepts of group actions both in the nonsingu-lar and and finite measure preserving cases, restricted infinite powers of probabilitymeasures, IDPFT actions and Poisson actions. In Section 2 we prove Theorems Band Corollary C. Section 3 is devoted to the proof of Theorems D and Corollary E.
1. Definitions and preliminaries
Nonsingular and measure preserving G -actions. Nonsingular actions appearin the proof of Theorem B. We remind several basic concepts related to them.
Definition 1.1.
Let S = ( S g ) g ∈ G be a nonsingular G -action on a standard prob-ability space ( Z, F , κ ).(i) S is called totally dissipative if the partition of Z into the S -orbits is measur-able and the S -stabilizer of a.e. point is compact, i.e. there is a measurablesubset of Z which meets a.e. S -orbit exactly once, and for a.e. z ∈ Z , thesubgroup { g ∈ G | S g z = z } is compact in G .(ii) S is called conservative if there is no any S -invariant subset A ⊂ Z ofpositive measure such that the restriction of S to A is totally dissipative.(iii) There is a unique (mod 0) partition of X into two invariant subsets D ( S )and C ( S ) such that S ↾ D ( S ) is totally dissipative and S ↾ D ( S ) is conser-vative. We call D ( S ) and C ( S ) the dissipative and conservative part of S respectively.(iv) S is called ergodic if each measurable S -invariant subset of Z is either µ -nullor µ -conull.(v) S is called weakly mixing if for each ergodic probability preserving G -action R = ( R g ) g ∈ G , the product G -action ( S g × R g ) g ∈ G is ergodic.(vi) S is called properly ergodic if it is ergodic and κ is not concentrated on asingle orbit.(vii) S is called sharply weak mixing [DaLe] if it is properly ergodic and for eachergodic conservative nonsingular G -action R = ( R g ) g ∈ G on a nonatomicprobability space, the product G -action ( S g × R g ) g ∈ G is either ergodic ortotally dissipative.We also remind some concepts related to finite measure preserving actions. Definition 1.2.
Suppose that κ ( Z ) = 1 and κ ◦ S g = κ for all g ∈ G .(i) S is called mixing if lim g →∞ κ ( S g A ∩ B ) = µ ( A ) µ ( B ) for all A, B ∈ F .3ii) A sequence of Borel subsets ( A n ) ∞ n =1 in X of strictly positive measure iscalled T -asymptotically invariant if for each compact subset K ⊂ G , wehave that sup g ∈ K κ ( A n △ T g A n ) → n → ∞ .(iii) T is called strongly ergodic if each T -asymptotically invariant sequence( A n ) ∞ n =1 is trivial, i.e. lim n →∞ κ ( A n )(1 − µ ( A n )) = 0.We now state a corollary from the Schmidt-Walters theorem [ScWa, Theo-rem 2.3]. Lemma 1.3.
Let S = ( S g ) g ∈ G be a mixing measure preserving action on a standardprobability space ( Y, C , ν ) . Then for each ergodic non-totally dissipative nonsingular G -action R = ( R g ) g ∈ G , the product G -action S × R := ( S g × R g ) g ∈ G is ergodic.Proof. We first note that a mixing action is properly ergodic. Hence if R = ( R g ) g ∈ G is properly ergodic then the claim of the proposition follows immediately from[ScWa, Theorem 2.3]. If R is not properly ergodic then there is a noncompactsubgroup H in G such that R is isomorphic to the G -action by left translations onthe coset space G/H endowed with a Haar measure. Hence S × R is ergodic if andonly if the H -action ( S ( h )) h ∈ H on ( Y, C , ν ) is ergodic. The later holds because S is mixing. (cid:3) Corollary 1.4.
Let S = ( S g ) g ∈ G be a mixing measure preserving action on astandard probability space ( Y, C , ν ) and let R = ( R g ) g ∈ G be a nonsingular G -actionon a standard probability space ( Z, D , κ ) . The following holds. (i) D ( S × R ) = Y × D ( R ) and C ( S × R ) = Y × C ( R ) . (ii) If R is conservative and F : Y × Z → C is an ( S × R ) -invariant Borelfunction then there exists a Borel R -invariant function f : Z → C such that F ( y, z ) = f ( z ) at a.e. ( y, z ) ∈ Y × Z .Proof. It suffices to consider the ergodic decomposition for R and apply Lemma 1.3to the product of S with each ergodic component of R . (cid:3) Restricted infinite powers of probability measures.
Let ( Y, C , γ ) be a stan-dard non-atomic probability space. Fix a sequence B := ( B n ) ∞ n =1 of subsets from C of strictly positive measure. Let ( X, B ) := ( Y, C ) ⊗ N . For each n ∈ N , we set B n := Y n × B n +1 × B n +2 × · · · ∈ B . Then B ⊂ B ⊂ · · · . We define a measure γ B on ( X, B ) by the following sequence of restrictions (see [Hi] for details): γ B ↾ B n := γγ ( B ) ⊗ · · · ⊗ γγ ( B n ) ⊗ γ ↾ B n +1 γ ( B n +1 ) ⊗ γ ↾ B n +2 γ ( B n +2 ) ⊗ · · · . Since the restrictions are compatible, γ B is well defined. We note that γ B issupported on the subset S ∞ n =1 B n ⊂ Y and γ B ( B n ) = Q nj =1 γ ( B j ) − for each n .Hence, γ B is σ -finite. It is infinite if and only if Q ∞ n =1 γ ( B n ) = 0. Definition 1.5.
We call γ B the restricted infinite power of γ with respect to B . Given a γ -preserving Borel bijection T of Y , we let T := N ∞ n =1 T and T B :=(
T B n ) ∞ n =1 . A straightforward verification shows that γ B ◦ T − = γ T B . Proposition 1.6. If P ∞ n =1 γ ( B n △ T B n ) γ ( B n ) < ∞ then T preserves γ B . roof. For each n ∈ N and an arbitrary Borel subset A ⊂ Y n , we let A ′ := A × B n +1 × B n +2 × · · · ∈ B . Then for every m > n ,( γ T B ↾ B m )( A ′ ) = γ ⊗ n ( A ) γ ( B ) · · · γ ( B n ) Y j>m γ ( B j ∩ T B j ) γ ( B j ) . Passing to the limit as m → ∞ , we obtain that γ T B ( A ′ ) = lim m →∞ ( γ B ↾ B m )( T A ′ ) = γ ⊗ n ( A ) γ ( B ) · · · γ ( B n ) = γ B ( A ′ ) . Hence γ B ◦ T − = γ T B = γ B , as desired. (cid:3) We note that under the condition of Proposition 1.6, γ B ( T B n ∩ B n ) = n Y j =1 γ ( B j ) − Y j>n γ ( B j ∩ T B j ) γ ( B j ) . Hence(1-1) lim n →∞ γ B ( T B n ∩ B n ) γ B ( B n ) = 1 . We note that the Hilbert space L ( X, γ B ) is the infinite tensor product of thesequence of Hilbert spaces ( L ( Y, γ ( B n ) − γ )) ∞ n =1 along the stabilizing sequence(1 B n ) ∞ n =1 of unit vectors (see [Gu] for the definition). In particular, we see that thelinear subspace ∞ [ n =1 ( f ⊗ O j>n B j (cid:12)(cid:12)(cid:12)(cid:12) f ∈ L ( Y n , γ ⊗ n ) ) is dense in L ( X, γ B ). Let U T and U T stand for the Koopman operators associatedto T and T in L ( Y, γ ) and L ( X, γ B ) respectively. The following proposition isverified straightforwardly. Lemma 1.7.
For each n ∈ N and arbitrary functions f, g ∈ L ( Y n , γ ⊗ n ) , we havethat U T (cid:18) f ⊗ O j>n B j (cid:19) = lim m →∞ ( U T ) ⊗ n f ⊗ (cid:18) m O j = n +1 T B j (cid:19) ⊗ O j>m B j and (cid:28) U T (cid:18) f ⊗ O j>n B j (cid:19) , g ⊗ O j>n B j (cid:29) = h ( U T ) ⊗ n f, g i Q nj =1 γ ( B j ) Y j>n γ ( T B j ∩ B j ) γ ( B j ) . IDPFT-actions.
IDPFT-actions were introduced in [DaLe] in the case, where G = Z . In [DaKo], IDPFT actions of arbitrary discrete countable groups werestudied. In this paper we consider IDPFT-actions for arbitrary locally compactsecond countable groups. 5 efinition 1.8. Let T n = ( T n ( g )) g ∈ G be an ergodic measure preserving G -actionon a standard probability space ( Y n , C n , ν n ), let µ n be a probability measure on C n and let µ n ∼ ν n for each n ∈ N . We put ( X, B , µ ) := N ∞ n =1 ( Y n , C n , µ n ), T ( g ) := N ∞ n =1 T n ( g ) and T := ( T ( g )) g ∈ G . If µ ◦ T ( g ) ∼ µ for each g ∈ G thenthe nonsingular dynamical system ( X, B , µ, T ) is called an infinite direct productof finite types (IDPFT) .We will need the following fact, extending [DaLe, Proposition 2.3] from Z -actionsto arbitrary G -actions. Proposition 1.9.
Let ( X, B , µ, T ) be an IDPFT system as in Definition 1.8. If T n is mixing for each n ∈ N then T is either totally dissipative or conservative. If T is conservative then T is sharply weak mixing.Proof. It follows from Corollary 1.4(i) that D ( T ) = Y × · · · × Y n × D (cid:18) O j>n T j (cid:19) mod 0for each n >
0. By Kolmogorov’s 0-1 law that either µ ( D ( T )) = 0 and hence T isconservative or µ ( D ( T )) = 1 and hence T is totally dissipative. Thus, the first claimis proved. We do not provide a proof for the second claim because it is an almostverbal repetition of the proof of [DaLe, Proposition 2.3]: just replace the referenceto [DaLe, Theorem B] there with a reference to Corollary 1.4(ii). Of course, µ isnot concentrated on a single orbit. (cid:3) Poisson suspensions (see [CoFoSi] and [Ro] for details) . Let ( X, B ) be a standardBorel space and let µ be an infinite σ -finite non-atomic measure on X . Let X ∗ bethe set of purely atomic ( σ -finite) measures on X . For each subset A ∈ B with0 < µ ( A ) < ∞ , we define a mapping N A : X ∗ → R by setting N A ( ω ) := ω ( A ).Let B ∗ stand for the smallest σ -algebra on X ∗ such that the mappings N A are all B ∗ -measurable. There is a unique probability measure µ ∗ on ( X ∗ , B ∗ ) satisfyingthe following two conditions:— the measure µ ∗ ◦ N − A is the Poisson distribution with parameter µ ( A ) foreach A ∈ B with ∞ > µ ( A ) > A , . . . , A q of mutually disjoint subsets A , . . . , A q ∈ B of finite positive measure, the corresponding random variables N A , . . . , N A q defined on the space ( X ∗ , B ∗ , µ ∗ ) are independent.Then ( X ∗ , B ∗ , µ ∗ ) is a Lebesgue space. For each µ -preserving G -action T =( T g ) g ∈ G , we define a G -action T ∗ = ( T ∗ g ) g ∈ G on ( X ∗ , B ∗ , µ ∗ ) by setting T ∗ g ω := ω ◦ T − g for all ω ∈ X ∗ and g ∈ G .Then T ∗ preserves µ ∗ . Definition 1.10.
The dynamical system ( X ∗ , B ∗ , µ ∗ , T ∗ ) is called the Poissonsuspension of ( X, B , µ, T ). A probability preserving G -action is called Poisson if itis isomorphic to a Poisson suspension of some infinite σ -finite measure preserving G -action. 6 nitary representations of G and Koopman representations of measurepreserving actions. Let V = ( V ( g )) g ∈ G be a weakly continuous unitary repre-sentation of G in a separable Hilbert space H . We will always assume that V is acomplexification of an orthogonal representation of G in a real Hilbert space. Definition 1.11. V is called weakly mixing if V has no nontrivial finite dimensionalinvariant subspaces.The Fock space F ( H ) over H is the orthogonal sum L ∞ n =0 H ⊙ n , where H ⊙ n isthe n -th symmetric tensor power of H when n > H ⊙ := C . By exp V =(exp V ( g )) g ∈ G we denote the corresponding unitary representation of G in F ( H ),i.e. exp V ( g ) := L ∞ n =0 V ( g ) ⊙ n for each g ∈ G .Let T = ( T g ) g ∈ G be a measure preserving G -action on a σ -finite nonatomicstandard measure space ( X, B , µ ). Denote by U T = ( U T ( g )) g ∈ G the associated(weakly continuous) unitary Koopman representation of G in L ( X, µ ): U T ( g ) f := f ◦ T − g , for all g ∈ G. Let L ( X, µ ) := L ( X, µ ) ⊖ C = { f ∈ L ( X, µ ) | R X f dµ = 0 } . We will need thefollowing fact. Fact 1.12.
Let V and ( X, B , µ, T ) be as above in this subsesction. (i) If µ ( X ) < ∞ then T is weakly mixing if and only if U T ↾ L ( X, µ ) is weaklymixing [BeRo] . (ii) V is weakly mixing if and only if (exp V ) ↾ ( F ( H ) ⊖ C ) is weakly mixing [GlWe1, Theorem A3] . (iii) If µ ( X ) = ∞ then U T ∗ is canonically unitarily equivalent to exp U T [Ro] . (iv) If ( Y, C , ν, S ) denote the Gaussian dynamical system ( G -action) associatedwith V then U S is canonically unitarily equivalent to exp V [Gu] . (v) V is weakly mixing if and only if there is a sequence g n → ∞ in G suchthat V ( g n ) → weakly as n → ∞ [BeRo, Corollary 1.6, Theorem 1.9] .
2. The Haagerup property
In this section we prove Theorem B (and hence Theorem A) and Corollary C.Prior to this we state without proof a folklore proposition.
Proposition 2.1.
Let S = ( S g ) g ∈ G be a measure preserving G -action on an infinite σ -finite standard measure space ( Z, Z , κ ) . Let Z ⊂ Z stand for the ring of subsetsof finite measure. Let Z ⊂ Z ⊂ · · · be a sequence of subsets from Z such that S ∞ n =1 Z n = Z and for each n > , there is a finite partition P n of Z n into subsetsof equal measure satisfying the following: — P n +1 ≻ P n for each n , — W ∞ n =1 P n is dense in ( Z , κ ) , i.e. for each B ∈ Z and ǫ > , there is n > and a P n -measurable subset B n with κ ( B △ B n ) < ǫ . (i) If for each compact subset K ⊂ G , an integer n > and a P n -atom A ,there exist a finite family g , . . . , g l ∈ G \ K and mutually disjoint subsets A , . . . , A l of A such that F li =1 S g i A i ⊂ A and κ ( F i =1 A i ) > . κ ( A ) then S is conservative. (ii) If for each n > and every pair of P n -atoms A and B , there exist a finitefamily g , . . . , g l ∈ G and mutually disjoint subsets A , . . . , A l of A such hat F li =1 S g i A i ⊂ B and κ ( F i =1 A i ) > . κ ( A ) then S is ergodic. Hence T is weakly mixing.Proof of Theorem B. The implications (ii) ⇒ (i) and (iii) ⇒ (i) are trivial.We now prove (i) ⇒ (ii). Let G have the Haagerup property. By [Ch–Va, The-orem 2.2.2], there is a mixing measure preserving free G -action T = ( T g ) g ∈ G ona standard probability space ( Y, C , γ ) and a T -asymptotically invariant sequence B := ( B n ) ∞ n =1 such that γ ( B n ) = 0 . n ∈ N . Passing to a subsequence, ifnecessary, we may (and will) assume that for each compact subset K ⊂ G ,(2-1) ∞ X n =1 sup g ∈ K γ ( T g B n △ B n ) < + ∞ . We now let ( X, B ) := ( Y, C ) ⊗ N . Endow this standard Borel space with the ( σ -finite) restricted infinite power γ B of γ with respect to B . Since Q n ∈ N γ ( B n ) = 0,it follows that γ B ( X ) = ∞ . For g ∈ G , let T g := T ⊗ N g . In view of (2-1), itfollows from Proposition 1.6 that γ B ◦ T g = γ B . Thus, T := ( T g ) g ∈ G is a measurepreserving G -action on ( X, B , γ B ). Since the map X ∋ x = ( y n ) ∞ n =1 → y ∈ Y intertwines T with T and T is free, T is free too. For each n ∈ N , define a subset B n ⊂ X is the same way as in §
1. It follows from (1-1) that the sequence ( B n ) n ∈ N is T -Følner. Moreover, it is exhausting and γ B ( B n ) = 2 n for each n . To show that T is of 0-type, we first note that since T is mixing then for each pair of integers n < m , lim g →∞ m Y j = n γ ( T g B j ∩ B j ) γ ( B j ) = m Y j = n γ ( B j ) = 2 − m + n − . Hence given two functions f, r ∈ L ( Y n , γ ⊗ n ), we deduce from Lemma 1.7 thatlim g →∞ (cid:28) U T (cid:18) f ⊗ O j>n B j (cid:19) , g ⊗ O j>n B j (cid:29) = 0 . This implies that T is of 0-type. Our next claim is that upon a replacement of γ B with an equivalent probabilitymeasure, T is an IDPFT. For that, we choose a sequence of reals ( ǫ n ) ∞ n =1 such that0 < ǫ n < n and P ∞ n =1 ǫ n < ∞ . Then we define, for each n ∈ N , a Borelfunction φ n : Y → R + by setting φ n := 2 ǫ n X \ B n + 2(1 − ǫ n )1 B n . Since γ ( B n ) = , a simple verification yields that R Y φ n dγ = 1. Denote by µ n theprobability measure on Y such that µ n ∼ γ and dµ n dγ := φ n . We now recall thatgiven two probability measures α and β on ( Y, C ) such that α ≺ γ and β ≺ γ , thesquared Hellinger distance between α and β is H ( α, β ) := 12 Z Y s dαdγ − s dβdγ ! dγ. It is worthy to note that at this point we have proved completely Theorem A. H (cid:18) γ ( B n ) γ ↾ B n , µ n (cid:19) = 12 Z Y ( √ · B n − p φ n ) dγ = 12 √ √ − ǫ n − + ǫ n )and hence P ∞ n =1 H ( γ ( B n ) γ ↾ B n , µ n ) ≤ + ∞ . Therefore, by [Hi, Theorems 3.9,3.6], γ B ∼ N ∞ n =1 µ n . Hence T is an IDPFT, as claimed.We now deduce from Proposition 1.9 that T is either sharply weak mixing ortotally dissipative. Therefore, to complete the proof of (i) ⇒ (ii), it suffices to showthat there exists a subsequence of B such that the corresponding G -action T (de-termined by this subsequence of B ) is conservative. Let ( P n ) ∞ n =1 be a refiningsequence of finite partitions of Y into Borel subsets of equal measure such that W ∞ n =1 P n is dense in ( C , γ ) and B n is P n -measurable for each n ∈ N . We set P ′ n := { P × B n +1 × B n +2 × · · · | P ∈ ( P n ) ⊗ n } . Then P ′ n is a finite partition of B n into subsets of equal measure for each n ∈ N . Moreover, P ′ ≺ P ′ ≺ · · · and W ∞ n =1 P ′ n is dense in ( B , γ B ). Fix a sequence ( K n ) ∞ n =1 of compact subsets of G such that K ⊂ K ⊂ · · · and S ∞ n =1 K n = G . Then for every n >
0, there is afinite subset F n ⊂ G \ K n such that for each atom P ∈ ( P n ) ⊗ n there is a family ofmeasured subsets ( P f ) f ∈ F n of P satisfying the following conditions:— P f ∩ P h = ∅ and ( T f ) ⊗ n P f ∩ ( T h ) ⊗ n P h = ∅ if f = h ,— F f ∈ F ( T f ) ⊗ n P f ⊂ P and— γ ⊗ n ( F f ∈ F n P f ) > . γ ⊗ n ( P ).Passing to a subsequence in B = ( B n ) ∞ n =1 we may assume without loss of generalitythat max f ∈ F n Q k>n γ ( T f B k ∩ B k ) γ ( B k ) > . n . Take an atom P ′ ∈ P ′ n for some n ∈ N . Then P ′ = P × B n +1 × B n +2 × · · · for an atom P ∈ ( P n ) ⊗ n . We nowset P ′ f := P f × ( B n +1 ∩ T − f B n +1 ) × ( B n +2 ∩ T − f B n +2 ) × · · · for each f ∈ F n .Then ( P ′ f ) f ∈ F n are mutually disjoint Borel subsets of P ′ and ( T f P ′ f ) f ∈ F n are alsomutually disjoint Borel subsets of P ′ . Moreover,(2-2) γ B (cid:18) G f ∈ F n P ′ f (cid:19) = X f ∈ F n γ ⊗ n ( P f ) Q nj =1 γ ( B j ) Y k>n γ ( T f B k ∩ B k ) γ ( B k ) > X f ∈ F n γ ⊗ n ( P f ) Q nj =1 γ ( B j )= γ ⊗ n ( F f ∈ F n P f )2 Q nj =1 γ ( B j ) > γ B ( P )4 . It follows from this and Proposition 2.1(i) that T is conservative, as desired.To prove (1) ⇒ (3), consider the dynamical system ( X, B , γ B , T ) constructedabove. Since T is not strongly ergodic, we have that for each α ∈ (0 , T -asymptotically invariant sequence ( A n ) ∞ n =1 such that γ ( A n ) = α for each n ∈ N . Hence, using the standard diagonalization argument, we can select anasymptotically invariant sequence ( A n ) ∞ n =1 such that γ ( A n ) = Q nj =1 γ ( B j ) for each9 ∈ N . We now set A n := Y n − × A n × B n +1 × B n +2 × · · · ⊂ X. Then γ B ( A n ) = γ ( A n ) Q nj =1 γ ( B j ) = 1 and γ B ( T g A n ∩ A n ) = n − Y j =1 γ ( B j ) − ! γ ( A n ∩ T g A n ) γ ( B n ) Y j>n γ ( B j ∩ T g B j ) γ ( B j ) . Hence for a compact subset K ⊂ G , we obtain that sup g ∈ K γ B ( T g A n ∩ A n ) → n → ∞ , as desired. (cid:3) Proof of Corollary C.
The “if” part follows from [Ch–Va, Theorem 2.2.2]. Weprove the “only if” part. Let G has the Haagerup property. Consider the dynamicalsystem ( X, B , γ B , T ) constructed in the proof of Theorem B. Let ( A n ) ∞ n =1 be the T -Følner sequence such that γ B ( A n ) = 1 for each n ∈ N . Denote by ( X ∗ , ( γ B ) ∗ , T ∗ )the Poisson suspension of ( X, γ B , T ). For each n ∈ N , we set [ A n ] := { ω ∈ X ∗ | ω ( A n ) = 0 } . Then ( γ B ) ∗ ([ A n ] ) = e − γ B ( A n ) = e − and ( γ B ) ∗ ( T ∗ g [ A n ] ∩ [ A n ] ) = ( γ B ) ∗ ([ T g A n ] ∩ [ A n ] )= ( γ B ) ∗ ([ T g A n ∪ A n ] )= e − γ B ( T g A n ∪ A n ) . Since for each compact subset K ⊂ G ,sup g ∈ K | γ B ( T g A n ∪ A n ) − γ B ( A n ) | → , we obtain that sup g ∈ K | ( γ B ) ∗ T ∗ g [ A n ] ∩ [ A n ] ) − ( γ B ) ∗ ([ A n ] ) | → n → ∞ . Thus, the sequence ([ A n ] ) ∞ n =1 is nontrivial and T ∗ -asymptoticallyinvariant. Thus T ∗ is not strongly ergodic. Since T is of 0-type, it follows fromFact 1.12(iii) that U T ∗ ( g ) ↾ L (cid:0) X ∗ , ( γ B ) ∗ (cid:1) → g → ∞ . Hence T ∗ ismixing. (cid:3)
3. Pairs of groups with Kazhdan property (T)
In this section we prove Theorem D and Corollary E. Prior to we note that if S be a weakly mixing measure preserving G -action on an infinite σ -finite standardmeasure space ( Y, F , ν ) then the Koopman representation U S is weakly mixing.Indeed, suppose that U S contains a finite dimensional subspace. Then there isa unitary representation V of G in a finite dimensional Hilbert space H and anontrivial mapping F : Y → H such F ( S g y ) = V ( g ) F ( y ) for each g ∈ G ata.e. y ∈ Y . By [GlWe2, Theorem 1.1], F is constant a.e. Since ν is infinite andthe mapping Y ∋ y
7→ h F ( y ) , h i belongs to L ( Y, ν ) for each h ∈ H , it follows that10 = 0. Therefore U S is weakly mixing, as claimed. It now follows from Fact 1.12(v)that S is of weak 0-type. Therefore Theorem D implies [Jo2, Theorem 1.5]. Proof of Theorem D. (i) If there is an S -Følner sequence then ( ∗ ) holds for theKoopman representation U S of G . Hence the restriction of U S to H has a nontrivialinvariant vector. Hence the action S ↾ H has an invariant subset of positive finitemeasure. This contradicts to the condition of (i).(ii) Let H ⊂ G do not have property (T). The beginning of our argument isa slight modification of the proof of [Jo2, Theorem 5]. There exists a condi-tionally negative definite function ψ : G → R + which is unbounded on H [Jo,Theorem 1.2(a4’)]. By the Schoenberg theorem, for each t >
0, the function φ t := e − t − ψ : G → R + is positive definite. Hence the GNS-construction yieldsa triplet ( V t , H t , ξ t ), consisting of a separable Hilbert space H t , a unitary represen-tation V t of G in H and a V t -cyclic unit vector ξ t ∈ H t such that h V t ( g ) ξ t , ξ t i = φ t ( g )for each g ∈ G . Since ψ is unbounded on H , the restriction of V t to H is weaklymixing by Fact 1.12(v). Since φ t takes only real values, V t is the complexification ofan orthogonal representation of G . We will now argue as in the case (a) of the proofof the main result from [CoWe]. Let V := L ∞ n =1 V n . Of course, the restriction of V to H is weakly mixing. Moreover, V is also the complexification of an orthogonalrepresentation of G . Denote by T = ( T g ) g ∈ G the corresponding Gaussian measurepreserving G -action on a standard probability space ( Y, C , γ ). By Fact 1.12(iv),the associated Koopman representation U T of G is unitarily equivalent to exp V .Since exp( V ↾ H ) = (exp V ) ↾ H , it follows from Fact 1.12(ii) that the unitaryrepresentation ( U T ( h )) h ∈ H is weakly mixing on L ( Y, γ ). Hence ( T h ) h ∈ H is weaklymixing by Fact 1.12(i). Since H n is a subspace of H , we obtain that ξ n ∈ L ( Y, γ ), ξ n is a centered Gaussian variable on Y and h U T ( g ) ξ n , ξ n i = h V n ( g ) ξ n , ξ n i = φ n ( g ) = e − ψ ( g ) /n for each g ∈ G. Let B n := { y ∈ Y | ξ n ( y ) > } . Then µ ( B n ) = 0 . k ξ n k = 0 .
5. As in the proof ofthe main result from [CoWe], we obtain that µ ( T g B n △ B n ) = arccos h U T ( g ) ξ n , ξ n i π = arccos e − ψ ( g ) /n π . This yields that the sequence B := ( B n ) ∞ n =1 is S -asymptotically invariant. More-over, without loss of generality we may (and will) assume that (2-1) holds for B .Consider now the dynamical system ( X, B , γ B , T ) as in the proof of Theorem 2.2.Then γ B ( X ) = ∞ and T preserves γ B by Proposition 1.6. As was shown inthe proof of Theorem B, the sequence ( B n ) n ∈ N (defined there) is T -Følner andexhausting.We now show how to choose a subsequence in B in such a way that the restrictionof the corresponding T (we recall again that T is determined by B ) to H is weaklymixing. Let ( P n ) ∞ n =1 denote the same sequence of finite partitions as in the proofof Theorem B. Since ( T h ) h ∈ H is weakly mixing, the H -action ( T ⊗ nh ) h ∈ H on theprobability space ( X n , γ ⊗ n ) is ergodic. Hence for every n >
0, there is a subset F n ⊂ H such that for every two atoms P, Q of the partition ( P n ) ⊗ n of B n × B n ,there is a family of measured subsets P f of P such that— P f ∩ P h = ∅ and ( T f ) ⊗ n P f ∩ ( T f ) ⊗ n P f = ∅ if f = h ,— F f ∈ F ( T f ) ⊗ n P f ⊂ Q and— γ ⊗ n ( F f ∈ F n P f ) > . γ ⊗ n ( P ). 11e now set P ′ := P × ( B n +1 × B n +2 × · · · ) × ⊂ B n × B n ,Q ′ := Q × ( B n +1 × B n +2 × · · · ) × ⊂ B n × B n and P ′ f := P f × (cid:0) ( B n +1 ∩ T − f B n +1 ) × ( B n +2 ∩ T − f B n +2 ) × · · · (cid:1) × . Then P n := ( P ′ ) P ∈ ( P n ) ⊗ n is a finite partition of B n × B n into subsets of equalmeasure, P ′ ≺ P ′ ≺ · · · and W ∞ n =1 P ′ n is dense in ( B ⊗ B , γ B ⊗ γ B ). We alsonote that ( P ′ f ) f ∈ F n are mutually disjoint subsets of P ′ and (( T f × T f ) P ′ f ) f ∈ F n aremutually disjoint subsets of Q ′ . Arguing as in (2-2), we obtain that( γ B ⊗ γ B ) (cid:18) G f ∈ F n P ′ f (cid:19) >
18 ( γ B ⊗ γ B )( P ′ ) . It follows from that and Proposition 2.1(ii) that the H -action ( T g × T g ) g ∈ H isergodic. Hence T ↾ H is weakly mixing by [GlWe2, Theorem 1.1]. (cid:3) Corollary E follows from Theorem D in the same way as Corollary C followsfrom Theorem B.
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