哈格鲁普性质和哈萨克对通过遍历无限测度保持作用

IF 0.7 3区 数学 Q2 MATHEMATICS
A. I. Danilenko
{"title":"哈格鲁普性质和哈萨克对通过遍历无限测度保持作用","authors":"A. I. Danilenko","doi":"10.4064/sm210702-27-10","DOIUrl":null,"url":null,"abstract":"It is shown that a locally compact second countable group G has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free G-action T = (Tg)g∈G on an infinite σ-finite standard measure space (X, μ) admitting a T -Følner sequence (i.e. a sequence (An)∞n=1 of measured subsets of finite measure such that A1 ⊂ A2 ⊂ · · · , ⋃ ∞ n=1 An = X and limn→∞ supg∈K μ(TgAn△An) μ(An) = 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 refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint. 0. Introduction Throughout this paper G is a non-compact locally compact second countable group. It has the Haagerup property if there is a weakly continuous unitary representation V of G in a separable Hilbert space H such that limg→∞ V (g) = 0 in the weak operator topology and (∗) for each ǫ > 0 and every compact subset K ⊂ G, there is a unit vector ξ ∈ H such that supg∈G ‖V (g)ξ − ξ‖ < ǫ. Of course, the amenable groups have the Haagerup property. The class of discrete countable Haagerup groups contains the free groups and is closed under free products and wreath products [CoStVa]. For more information about the Haagerup property we refer to [Ch–Va]. There is a purely dynamical description of this property: G is Haagerup if and only if there exists a mixing non-strongly ergodic probability preserving free G-action [Ch–Va, Theorem 2.2.2] (see §1 for the definitions). Recently, an infinite measure preserving counterpart of this result was discovered in [DeJoZu]: Theorem A. G has the Haagerup property if and only if there is a 0-type measure preserving G-action T = (Tg)g∈G on an infinite σ-finite measure space (X,B, μ) admitting a sequence of non-negative unit vectors (ξn) ∞ n=1 in L (X,μ) such that limn→∞ supg∈K〈ξn ◦ Tg, ξn〉 = 1 for each compact K ⊂ G. We recall that T is called of 0-type if limg→∞ μ(TgA ∩ B) = 0 for all subsets A,B ∈ B of finite measure. In this paper we provide a much shorter alternative proof of Theorem A which is grounded on the Moore-Hill concept of restricted infinite products of probability measures [Hi]. We note that the 0-type for infinite measure preserving systems is a natural counterpart of the mixing for probability preserving systems. However unlike mixing, Typeset by AMS-TEX 1 the 0-type is not a “strong” asymptotic property. It implies neither weak mixing nor ergodicity. Moreover, the totally dissipative actions are all of 0-type. In view of that the description in Theorem A does not look sharp from the ergodic theory point of view. Our first main result in this work is the following finer ergodic criterion of the Haagerup property. Theorem B. The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space admitting an exhausting T -Følner sequence of subsets. (iii) There exists a a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space (X,B, μ) admitting a T -Følner sequence (An) ∞ n=1 such that μ(An) = 1 for all n ∈ N. We say that (An) ∞ n=1 is T -Følner if μ(An) <∞ and supg∈K μ(An△TgAn) μ(An) → 0 as n → ∞ for each compact subset K ⊂ G. If A1 ⊂ A2 ⊂ · · · and ⋃∞ n=1An = X , we say that (An) ∞ n=1 is exhausting. We note that sharp mixing (see §1 for the definition) implies ergodicity and weak mixing. To prove (the non-trivial part of) Theorem B, we apply the Moore-Hill construction [Hi] to the mixing non-strongly ergodic G-action from [Ch–Va, Theorem 2.2.2] (cf. the construction of II∞ ergodic Poisson suspensions of countable amenable groups from [DaKo]). Then we observe that the action T that we obtain is IDPFT (see §1 and [DaLe], where such actions were introduced). Hence, by the properties of IDPFT systems, T is sharply weak mixing whenever we show that it is conservative. To show the conservativeness of T is remains to choose the parameters of the Moore-Hill construction in an a appropriate way. As a corollary from Theorem B, we obtain one more dynamical characterization of 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) which is a reciprocal to the Haagerup property. We recall [Jo1, Definition 1.1] that given a non-compact closed subgroup H of G, the pair H ⊂ G has property (T) if for each unitary representation V of G satisfying (∗), there is a unit vector which is invariant under V (h) for every h ∈ H . Using the techniques developed for proving Theorem B we obtain an ergodic (non-spectral) characterization of Kazhdan pairs that refines a spectral characterization from [Jo2]. Theorem D. (i) If a pair H ⊂ G has property (T) then each measure preserving G-action S = (Sg)g∈G on a σ-finite infinite standard measure space (Y,C, ν), such that S ↾ H := (Sh)h∈H has no invariant subsets of positive finite measure, admits no S-Følner sequences. 1Conservativeness, ergodicity, weak mixing and sharp weak mixing are not spectral invariants of the underlying dynamical systems. Hence the principal difference of Theorem B from Theorem A is that it provides non-spectral ergodic characterization of the Haagerup property. 2 (ii) If a pair H ⊂ G does not have property (T) then there is a measure preserving 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 hn → ∞ in H such that limn→∞ ν(ShnA ∩ B) = 0 for all subsets A,B ∈ C of finite measure. Then replacing the “has no invariant subsets of positive finite measure” in (i) with a stronger “is of weak 0-type”, and the “weakly mixing” in (ii) with a weaker “of weak 0-type” we obtain exactly [Jo2, Theorem 1.5]. Corollary E. A pair H ⊂ G has property (T) if and only if every (probability preserving) 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 definitions related to the basic dynamical concepts of group actions both in the nonsingular and and finite measure preserving cases, restricted infinite powers of probability measures, IDPFT actions and Poisson actions. In Section 2 we prove Theorems B and 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 appear in the proof of Theorem B. We remind several basic concepts related to them. Definition 1.1. Let S = (Sg)g∈G be a nonsingular G-action on a standard probability space (Z,F, κ). (i) S is called totally dissipative if the partition of Z into the S-orbits is measurable and the S-stabilizer of a.e. point is compact, i.e. there is a measurable subset of Z which meets a.e. S-orbit exactly once, and for a.e. z ∈ Z, the subgroup {g ∈ G | Sgz = z} is compact in G. (ii) S is called conservative if there is no any S-invariant subset A ⊂ Z of positive 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 conservative. 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-null or μ-conull. (v) S is called weakly mixing if for each ergodic probability preserving G-action R = (Rg)g∈G, the product G-action (Sg ×Rg)g∈G is ergodic. (vi) S is called properly ergodic if it is ergodic and κ is not concentrated on a single orbit. (vii) S is called sharply weak mixing [DaLe] if it is properly ergodic and for each ergodic conservative nonsingular G-action R = (Rg)g∈G on a nonatomic probability space, the product G-action (Sg × Rg)g∈G is either ergodic or totally dissipative. We also remind some concepts related to finite measure preserving actions. Definition 1.2. Suppose that κ(Z) = 1 and κ ◦ Sg = κ for all g ∈ G. (i) S is called mixing if limg→∞ κ(SgA ∩B) = μ(A)μ(B) for all A,B ∈ F. 3 (ii) A sequence of Borel subsets (An) ∞ n=1 in X of strictly positive measure is called T -asymptotically invariant if for each compact subset K ⊂ G, we have that supg∈K κ(An△TgAn) → 0 as n→ ∞. (iii) T is called strongly ergodic if each T -asymptotically invariant sequence (An) ∞ n=1 is trivial, i.e. limn→∞ κ(An)(1− μ(An)) = 0. We now state a corollary from the Schmidt-Walters theorem [ScWa, Theorem 2.3]. Lemma 1.3. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν). Then for each ergodic non-totally dissipative nonsingular G-action R = (Rg)g∈G, the product G-action S ×R := (Sg ×Rg)g∈G is ergodic. Proof. We first note that a mixing action is properly ergodic. Hence if R = (Rg)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 noncompact subgroup H in G such that R is isomorphic to the G-action by left translations on the coset space G/H endowed with a Haar measure. Hence S ×R is ergodic if and only if the H-action (S(h))h∈H on (Y,C, ν) is ergodic. The later holds because S is mixing. Corollary 1.4. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν) and let R = (Rg)g∈G be a nonsingular G-action on 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)-in","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Haagerup property and Kazhdan pairs\\nvia ergodic infinite measure preserving actions\",\"authors\":\"A. I. Danilenko\",\"doi\":\"10.4064/sm210702-27-10\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is shown that a locally compact second countable group G has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free G-action T = (Tg)g∈G on an infinite σ-finite standard measure space (X, μ) admitting a T -Følner sequence (i.e. a sequence (An)∞n=1 of measured subsets of finite measure such that A1 ⊂ A2 ⊂ · · · , ⋃ ∞ n=1 An = X and limn→∞ supg∈K μ(TgAn△An) μ(An) = 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 refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint. 0. Introduction Throughout this paper G is a non-compact locally compact second countable group. It has the Haagerup property if there is a weakly continuous unitary representation V of G in a separable Hilbert space H such that limg→∞ V (g) = 0 in the weak operator topology and (∗) for each ǫ > 0 and every compact subset K ⊂ G, there is a unit vector ξ ∈ H such that supg∈G ‖V (g)ξ − ξ‖ < ǫ. Of course, the amenable groups have the Haagerup property. The class of discrete countable Haagerup groups contains the free groups and is closed under free products and wreath products [CoStVa]. For more information about the Haagerup property we refer to [Ch–Va]. There is a purely dynamical description of this property: G is Haagerup if and only if there exists a mixing non-strongly ergodic probability preserving free G-action [Ch–Va, Theorem 2.2.2] (see §1 for the definitions). Recently, an infinite measure preserving counterpart of this result was discovered in [DeJoZu]: Theorem A. G has the Haagerup property if and only if there is a 0-type measure preserving G-action T = (Tg)g∈G on an infinite σ-finite measure space (X,B, μ) admitting a sequence of non-negative unit vectors (ξn) ∞ n=1 in L (X,μ) such that limn→∞ supg∈K〈ξn ◦ Tg, ξn〉 = 1 for each compact K ⊂ G. We recall that T is called of 0-type if limg→∞ μ(TgA ∩ B) = 0 for all subsets A,B ∈ B of finite measure. In this paper we provide a much shorter alternative proof of Theorem A which is grounded on the Moore-Hill concept of restricted infinite products of probability measures [Hi]. We note that the 0-type for infinite measure preserving systems is a natural counterpart of the mixing for probability preserving systems. However unlike mixing, Typeset by AMS-TEX 1 the 0-type is not a “strong” asymptotic property. It implies neither weak mixing nor ergodicity. Moreover, the totally dissipative actions are all of 0-type. In view of that the description in Theorem A does not look sharp from the ergodic theory point of view. Our first main result in this work is the following finer ergodic criterion of the Haagerup property. Theorem B. The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space admitting an exhausting T -Følner sequence of subsets. (iii) There exists a a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space (X,B, μ) admitting a T -Følner sequence (An) ∞ n=1 such that μ(An) = 1 for all n ∈ N. We say that (An) ∞ n=1 is T -Følner if μ(An) <∞ and supg∈K μ(An△TgAn) μ(An) → 0 as n → ∞ for each compact subset K ⊂ G. If A1 ⊂ A2 ⊂ · · · and ⋃∞ n=1An = X , we say that (An) ∞ n=1 is exhausting. We note that sharp mixing (see §1 for the definition) implies ergodicity and weak mixing. To prove (the non-trivial part of) Theorem B, we apply the Moore-Hill construction [Hi] to the mixing non-strongly ergodic G-action from [Ch–Va, Theorem 2.2.2] (cf. the construction of II∞ ergodic Poisson suspensions of countable amenable groups from [DaKo]). Then we observe that the action T that we obtain is IDPFT (see §1 and [DaLe], where such actions were introduced). Hence, by the properties of IDPFT systems, T is sharply weak mixing whenever we show that it is conservative. To show the conservativeness of T is remains to choose the parameters of the Moore-Hill construction in an a appropriate way. As a corollary from Theorem B, we obtain one more dynamical characterization of 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) which is a reciprocal to the Haagerup property. We recall [Jo1, Definition 1.1] that given a non-compact closed subgroup H of G, the pair H ⊂ G has property (T) if for each unitary representation V of G satisfying (∗), there is a unit vector which is invariant under V (h) for every h ∈ H . Using the techniques developed for proving Theorem B we obtain an ergodic (non-spectral) characterization of Kazhdan pairs that refines a spectral characterization from [Jo2]. Theorem D. (i) If a pair H ⊂ G has property (T) then each measure preserving G-action S = (Sg)g∈G on a σ-finite infinite standard measure space (Y,C, ν), such that S ↾ H := (Sh)h∈H has no invariant subsets of positive finite measure, admits no S-Følner sequences. 1Conservativeness, ergodicity, weak mixing and sharp weak mixing are not spectral invariants of the underlying dynamical systems. Hence the principal difference of Theorem B from Theorem A is that it provides non-spectral ergodic characterization of the Haagerup property. 2 (ii) If a pair H ⊂ G does not have property (T) then there is a measure preserving 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 hn → ∞ in H such that limn→∞ ν(ShnA ∩ B) = 0 for all subsets A,B ∈ C of finite measure. Then replacing the “has no invariant subsets of positive finite measure” in (i) with a stronger “is of weak 0-type”, and the “weakly mixing” in (ii) with a weaker “of weak 0-type” we obtain exactly [Jo2, Theorem 1.5]. Corollary E. A pair H ⊂ G has property (T) if and only if every (probability preserving) 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 definitions related to the basic dynamical concepts of group actions both in the nonsingular and and finite measure preserving cases, restricted infinite powers of probability measures, IDPFT actions and Poisson actions. In Section 2 we prove Theorems B and 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 appear in the proof of Theorem B. We remind several basic concepts related to them. Definition 1.1. Let S = (Sg)g∈G be a nonsingular G-action on a standard probability space (Z,F, κ). (i) S is called totally dissipative if the partition of Z into the S-orbits is measurable and the S-stabilizer of a.e. point is compact, i.e. there is a measurable subset of Z which meets a.e. S-orbit exactly once, and for a.e. z ∈ Z, the subgroup {g ∈ G | Sgz = z} is compact in G. (ii) S is called conservative if there is no any S-invariant subset A ⊂ Z of positive 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 conservative. 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-null or μ-conull. (v) S is called weakly mixing if for each ergodic probability preserving G-action R = (Rg)g∈G, the product G-action (Sg ×Rg)g∈G is ergodic. (vi) S is called properly ergodic if it is ergodic and κ is not concentrated on a single orbit. (vii) S is called sharply weak mixing [DaLe] if it is properly ergodic and for each ergodic conservative nonsingular G-action R = (Rg)g∈G on a nonatomic probability space, the product G-action (Sg × Rg)g∈G is either ergodic or totally dissipative. We also remind some concepts related to finite measure preserving actions. Definition 1.2. Suppose that κ(Z) = 1 and κ ◦ Sg = κ for all g ∈ G. (i) S is called mixing if limg→∞ κ(SgA ∩B) = μ(A)μ(B) for all A,B ∈ F. 3 (ii) A sequence of Borel subsets (An) ∞ n=1 in X of strictly positive measure is called T -asymptotically invariant if for each compact subset K ⊂ G, we have that supg∈K κ(An△TgAn) → 0 as n→ ∞. (iii) T is called strongly ergodic if each T -asymptotically invariant sequence (An) ∞ n=1 is trivial, i.e. limn→∞ κ(An)(1− μ(An)) = 0. We now state a corollary from the Schmidt-Walters theorem [ScWa, Theorem 2.3]. Lemma 1.3. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν). Then for each ergodic non-totally dissipative nonsingular G-action R = (Rg)g∈G, the product G-action S ×R := (Sg ×Rg)g∈G is ergodic. Proof. We first note that a mixing action is properly ergodic. Hence if R = (Rg)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 noncompact subgroup H in G such that R is isomorphic to the G-action by left translations on the coset space G/H endowed with a Haar measure. Hence S ×R is ergodic if and only if the H-action (S(h))h∈H on (Y,C, ν) is ergodic. The later holds because S is mixing. Corollary 1.4. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν) and let R = (Rg)g∈G be a nonsingular G-action on a standard probability space (Z,D, κ). The following holds. (i) D(S ×R) = Y ×D(R) and C(S ×R) = Y × C(R). 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引用次数: 2

摘要

使用为证明定理B而开发的技术,我们获得了Kazhdan对的遍历(非谱)特征,该特征从[Jo2]中细化了谱特征。定理D.(i)如果一对H⊂G具有性质(T),则在σ-有限无限标准测度空间(Y,C,Γ)上每个保持G-作用的测度S=(Sg)G∈G,使得S↾ H:=(Sh)H∈H不存在正有限测度的不变子集,不存在S-Følner序列。1保守性、遍历性、弱混合和锐弱混合不是底层动力系统的谱不变量。因此,定理B与定理A的主要区别在于,它提供了Haagerup性质的非谱遍历特征。2(ii)如果对H⊂G不具有性质(T),则在具有耗尽S-Følner序列的σ-有限无限测度空间上存在一个保测度的G-作用S↾ H是弱混合的。让我们说S↾ 如果存在子序列hn,则H为弱0-型→ ∞ 在H中,使得limn→∞ 对于有限测度的所有子集A,B∈C,Γ(ShnAåB)=0。然后,将(i)中的“正有限测度的无不变子集”替换为更强的“弱0-型”,将(ii)中的”弱混合“替换为较弱的”弱0-型“,我们精确地得到了[Jo2,定理1.5]。推论E.一对H⊂G具有性质(T)当且仅当具有弱混合H-子作用的每个(保概率)Poisson G-作用都是强遍历的。用“遍历”代替“弱混合”也是如此。论文概要如下。在第1节中,我们陈述了与群作用的基本动力学概念相关的所有必要定义,包括在非奇异和有限保测度的情况下,概率测度的受限无穷幂,IDPFT作用和泊松作用。在第2节中,我们证明了定理B和推论C。第3节专门讨论了定理D和推论E.1的证明。定义和预备条件非奇异和保测度G-作用。非奇异作用出现在定理B的证明中。我们提醒几个与之相关的基本概念。定义1.1。设S=(Sg)g∈g是标准概率空间(Z,F,κ)上的非奇异g-作用。(i) 如果Z到S-轨道的划分是可测量的,并且a.e.点的S-稳定器是紧致的,即存在Z的可测量子集正好满足a.e.S-轨道一次,并且对于a.e.Z∈Z,子群{g∈g|Sgz=z}在g中是紧致的。(ii)如果不存在任何正测度的S-不变子集A⊂z,则S称为守恒的,使得S对A的限制是完全耗散的。(iii)存在X到两个不变子集D(S)和C(S)的唯一(mod 0)划分,使得S↾ D(S)是完全耗散的并且S↾ D(S)是保守的。我们分别称D(S)和C(S)为S的耗散部分和守恒部分。(iv)如果Z的每个可测量的S不变子集都是μ-零或μ-圆锥,则S称为遍历的。(v) S称为弱混合,如果对于每个保遍历概率的G-作用R=(Rg)G∈G,乘积G-作用(Sg×Rg)G∈G是遍历的。(vi)如果S是遍历的并且κ不集中在单个轨道上,则称其为适当遍历。(vii)S称为锐弱混合[DaLe],如果它是适当遍历的,并且对于非原子概率空间上的每个遍历保守非奇异G-作用R=(Rg)G∈G,乘积G-作用(Sg×Rg)G∈G要么遍历,要么全耗散。我们还提醒了一些与有限测度保持作用有关的概念。定义1.2。假设κ(Z)=1并且κ◦ 对于所有g∈g,Sg=κ。(i)S称为混合,如果limg→∞ 对于所有A,B∈F.3(ii)严格正测度的X中的Borel子集(An)∞n=1的序列称为T-渐近不变如果对于每个紧子集K⊂G,我们有supg∈Kκ(An△TgAn)→ 0作为n→ ∞. (iii)如果每个T-渐近不变序列(An)∞n=1是平凡的,则T称为强遍历的,即limn→∞ κ(An)(1−μ。我们现在陈述Schmidt-Walters定理[ScWa,定理2.3]的一个推论。引理1.3。设S=(Sg)g∈g是标准概率空间(Y,C,Γ)上的混合测度保持作用。则对于每个遍历的非全耗散非奇异G-作用R=(Rg)G∈G,乘积G-作用S×R:=(Sg×Rg)G∈G是遍历的。证据我们首先注意到混合作用是适当的遍历性。因此,如果R=(Rg)g∈g是适当遍历的,则命题的声明立即从[ScWa,定理2.3]得出。如果R不是适当遍历的则g中存在一个非紧子群H,使得R通过在具有Haar测度的陪集空间g/H上的左平移同构于g-作用。因此,S×R是遍历的,当且仅当H-作用(S(H))H∈H在(Y,C,Γ)上是遍历的。后者成立是因为S在混合。推论1.4。 设S=(Sg)g∈g是标准概率空间(Y,C,Γ)上的混合测度保持作用,设R=(Rg)g≠g是一个标准概率空间上的非奇异g-作用。以下内容成立。(i) D(R)和C(S×R)=Y×。(ii)如果R是保守的并且F:Y×Z→ C是(S×R)-in
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions
It is shown that a locally compact second countable group G has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free G-action T = (Tg)g∈G on an infinite σ-finite standard measure space (X, μ) admitting a T -Følner sequence (i.e. a sequence (An)∞n=1 of measured subsets of finite measure such that A1 ⊂ A2 ⊂ · · · , ⋃ ∞ n=1 An = X and limn→∞ supg∈K μ(TgAn△An) μ(An) = 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 refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint. 0. Introduction Throughout this paper G is a non-compact locally compact second countable group. It has the Haagerup property if there is a weakly continuous unitary representation V of G in a separable Hilbert space H such that limg→∞ V (g) = 0 in the weak operator topology and (∗) for each ǫ > 0 and every compact subset K ⊂ G, there is a unit vector ξ ∈ H such that supg∈G ‖V (g)ξ − ξ‖ < ǫ. Of course, the amenable groups have the Haagerup property. The class of discrete countable Haagerup groups contains the free groups and is closed under free products and wreath products [CoStVa]. For more information about the Haagerup property we refer to [Ch–Va]. There is a purely dynamical description of this property: G is Haagerup if and only if there exists a mixing non-strongly ergodic probability preserving free G-action [Ch–Va, Theorem 2.2.2] (see §1 for the definitions). Recently, an infinite measure preserving counterpart of this result was discovered in [DeJoZu]: Theorem A. G has the Haagerup property if and only if there is a 0-type measure preserving G-action T = (Tg)g∈G on an infinite σ-finite measure space (X,B, μ) admitting a sequence of non-negative unit vectors (ξn) ∞ n=1 in L (X,μ) such that limn→∞ supg∈K〈ξn ◦ Tg, ξn〉 = 1 for each compact K ⊂ G. We recall that T is called of 0-type if limg→∞ μ(TgA ∩ B) = 0 for all subsets A,B ∈ B of finite measure. In this paper we provide a much shorter alternative proof of Theorem A which is grounded on the Moore-Hill concept of restricted infinite products of probability measures [Hi]. We note that the 0-type for infinite measure preserving systems is a natural counterpart of the mixing for probability preserving systems. However unlike mixing, Typeset by AMS-TEX 1 the 0-type is not a “strong” asymptotic property. It implies neither weak mixing nor ergodicity. Moreover, the totally dissipative actions are all of 0-type. In view of that the description in Theorem A does not look sharp from the ergodic theory point of view. Our first main result in this work is the following finer ergodic criterion of the Haagerup property. Theorem B. The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space admitting an exhausting T -Følner sequence of subsets. (iii) There exists a a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space (X,B, μ) admitting a T -Følner sequence (An) ∞ n=1 such that μ(An) = 1 for all n ∈ N. We say that (An) ∞ n=1 is T -Følner if μ(An) <∞ and supg∈K μ(An△TgAn) μ(An) → 0 as n → ∞ for each compact subset K ⊂ G. If A1 ⊂ A2 ⊂ · · · and ⋃∞ n=1An = X , we say that (An) ∞ n=1 is exhausting. We note that sharp mixing (see §1 for the definition) implies ergodicity and weak mixing. To prove (the non-trivial part of) Theorem B, we apply the Moore-Hill construction [Hi] to the mixing non-strongly ergodic G-action from [Ch–Va, Theorem 2.2.2] (cf. the construction of II∞ ergodic Poisson suspensions of countable amenable groups from [DaKo]). Then we observe that the action T that we obtain is IDPFT (see §1 and [DaLe], where such actions were introduced). Hence, by the properties of IDPFT systems, T is sharply weak mixing whenever we show that it is conservative. To show the conservativeness of T is remains to choose the parameters of the Moore-Hill construction in an a appropriate way. As a corollary from Theorem B, we obtain one more dynamical characterization of 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) which is a reciprocal to the Haagerup property. We recall [Jo1, Definition 1.1] that given a non-compact closed subgroup H of G, the pair H ⊂ G has property (T) if for each unitary representation V of G satisfying (∗), there is a unit vector which is invariant under V (h) for every h ∈ H . Using the techniques developed for proving Theorem B we obtain an ergodic (non-spectral) characterization of Kazhdan pairs that refines a spectral characterization from [Jo2]. Theorem D. (i) If a pair H ⊂ G has property (T) then each measure preserving G-action S = (Sg)g∈G on a σ-finite infinite standard measure space (Y,C, ν), such that S ↾ H := (Sh)h∈H has no invariant subsets of positive finite measure, admits no S-Følner sequences. 1Conservativeness, ergodicity, weak mixing and sharp weak mixing are not spectral invariants of the underlying dynamical systems. Hence the principal difference of Theorem B from Theorem A is that it provides non-spectral ergodic characterization of the Haagerup property. 2 (ii) If a pair H ⊂ G does not have property (T) then there is a measure preserving 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 hn → ∞ in H such that limn→∞ ν(ShnA ∩ B) = 0 for all subsets A,B ∈ C of finite measure. Then replacing the “has no invariant subsets of positive finite measure” in (i) with a stronger “is of weak 0-type”, and the “weakly mixing” in (ii) with a weaker “of weak 0-type” we obtain exactly [Jo2, Theorem 1.5]. Corollary E. A pair H ⊂ G has property (T) if and only if every (probability preserving) 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 definitions related to the basic dynamical concepts of group actions both in the nonsingular and and finite measure preserving cases, restricted infinite powers of probability measures, IDPFT actions and Poisson actions. In Section 2 we prove Theorems B and 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 appear in the proof of Theorem B. We remind several basic concepts related to them. Definition 1.1. Let S = (Sg)g∈G be a nonsingular G-action on a standard probability space (Z,F, κ). (i) S is called totally dissipative if the partition of Z into the S-orbits is measurable and the S-stabilizer of a.e. point is compact, i.e. there is a measurable subset of Z which meets a.e. S-orbit exactly once, and for a.e. z ∈ Z, the subgroup {g ∈ G | Sgz = z} is compact in G. (ii) S is called conservative if there is no any S-invariant subset A ⊂ Z of positive 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 conservative. 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-null or μ-conull. (v) S is called weakly mixing if for each ergodic probability preserving G-action R = (Rg)g∈G, the product G-action (Sg ×Rg)g∈G is ergodic. (vi) S is called properly ergodic if it is ergodic and κ is not concentrated on a single orbit. (vii) S is called sharply weak mixing [DaLe] if it is properly ergodic and for each ergodic conservative nonsingular G-action R = (Rg)g∈G on a nonatomic probability space, the product G-action (Sg × Rg)g∈G is either ergodic or totally dissipative. We also remind some concepts related to finite measure preserving actions. Definition 1.2. Suppose that κ(Z) = 1 and κ ◦ Sg = κ for all g ∈ G. (i) S is called mixing if limg→∞ κ(SgA ∩B) = μ(A)μ(B) for all A,B ∈ F. 3 (ii) A sequence of Borel subsets (An) ∞ n=1 in X of strictly positive measure is called T -asymptotically invariant if for each compact subset K ⊂ G, we have that supg∈K κ(An△TgAn) → 0 as n→ ∞. (iii) T is called strongly ergodic if each T -asymptotically invariant sequence (An) ∞ n=1 is trivial, i.e. limn→∞ κ(An)(1− μ(An)) = 0. We now state a corollary from the Schmidt-Walters theorem [ScWa, Theorem 2.3]. Lemma 1.3. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν). Then for each ergodic non-totally dissipative nonsingular G-action R = (Rg)g∈G, the product G-action S ×R := (Sg ×Rg)g∈G is ergodic. Proof. We first note that a mixing action is properly ergodic. Hence if R = (Rg)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 noncompact subgroup H in G such that R is isomorphic to the G-action by left translations on the coset space G/H endowed with a Haar measure. Hence S ×R is ergodic if and only if the H-action (S(h))h∈H on (Y,C, ν) is ergodic. The later holds because S is mixing. Corollary 1.4. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν) and let R = (Rg)g∈G be a nonsingular G-action on 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)-in
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来源期刊
Studia Mathematica
Studia Mathematica 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
72
审稿时长
5 months
期刊介绍: The journal publishes original papers in English, French, German and Russian, mainly in functional analysis, abstract methods of mathematical analysis and probability theory.
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