Krieger’s type for ergodic non-singular Poisson actions of non-(T) locally compact groups

Abstract

It is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger type. These actions are amenable if and only if G is amenable. If G has the Haagerup property, then (and only then) these actions can be chosen of 0-type. If G is amenable, then G admits weakly mixing Bernoulli actions of arbitrary Krieger type.

1 Introduction

This article continues investigation of ergodic properties of non-singular Poisson suspensions initiated by Z. Kosloff, E. Roy and the present author in a series of works [DaKo, DaKoRo1, DaKoRo2]. In this paper, we deal with the class of locally compact second countable groups without Kazhdan property (T) and a subclass of these groups with the Haagerup property (or a-T-menable groups according to M. Gromov). For properties of these groups and their applications in group representations theory, geometric group theory, operator algebras, ergodic theory, the Baum–Connes conjecture etc., we refer to the books [BeHaVa, Ch–Va] and references therein.

Our purpose is to find all possible Krieger types for the ergodic conservative Poisson actions of groups from the aforementioned two classes. We recall some standard definitions. Let G be a locally compact second countable group.

• • A weakly continuous unitary representation $V=(V(g))_{g\in G}$ of G in a separable Hilbert space $\mathcal {H}$ has almost invariant vectors if, for each $\epsilon>0$ and every compact subset $K\subset G$ , there is a unit vector $\xi \in \mathcal {H}$ such that $\sup _{g\in K}\|V(g)\xi -\xi \|<\epsilon $ .

• • G has Kazhdan’s property (T) if, for each unitary representation of G that has almost invariant vectors, there is a non-zero invariant vector.

• • G has the Haagerup property if there is a unitary representation V of G with almost invariant vectors and such that $V(g)\to 0$ as $g\to \infty $ in the weak operator topology.

We note that each amenable group has the Haagerup property (consider the left regular representation of G) and no group with the Haagerup property has property (T).

As it turned out, our main results provide new dynamical criteria for the property (T) and the Haagerup property (see Theorems A and B below). Prior to stating them, we recall the famous dynamical criterion for the Kazhdan property due to A. Connes and B. Weiss: G is non-(T) if and only if there is a weakly mixing probability preserving G-action that is not strongly ergodic [CoWe]. Another criterion, in terms of infinite measure-preserving actions, is provided in [Jo]: G is non-(T) if and only if there is an infinite measure-preserving G-action whose Koopman representation is weakly mixing and admits almost invariant vectors. Similar criteria were proved for the Haagerup property. The following are equivalent.

• • G has the Haagerup property.

• • There is a mixing probability-preserving G-action that is not strongly ergodic [Ch–Va].

• • There is a 0-type infinite measure-preserving G-action whose Koopman representation admits almost invariant vectors [DeJoZu].

We now characterize the (T)-property and the Haagerup property in terms of non-singular Poisson actions.

Theorem A. Let $\mathcal {K}:= \{III_{\unicode{x3bb} }\mid 0\le \unicode{x3bb} \le 1\}\cup \{II_{\infty }\}$ . The following are equivalent.

1. (1) G does not have Kazhdan’s property (T).

2. (2) Given $K \!\in \mathcal {K}$ , there is an ergodic non-singular Poisson G-action of Krieger type K.

3. (3) For each $K\!\in \mathcal {K}$ , there is an ergodic non-singular Poisson G-action $T^{*}$ of Krieger type K with the following extra properties: $T^{*}$ is (essentially) free, of infinite ergodic index, non-strongly ergodic and IDPFT.

We recall that a nonsingular G-action is called IDPFT if it isomorphic to the direct product of countably many nonsingular G-actions each of which admits an equivalent invariant probability measure (see [DaLe] and Definition 2.15 below).

Theorem B. Let $\mathcal {K}$ be as in Theorem A. The following are equivalent.

1. (1) G is a non-compact group with the Haagerup property.

2. (2) There is an ergodic non-singular Poisson G-action that is of 0-type and non-strongly ergodic.

3. (3) For each $K\!\in \mathcal {K}$ , there is an ergodic non-singular Poisson G-action $T^{*}$ of Krieger type K with the following extra properties: $T^{*}$ is free, of infinite ergodic index, of 0-type, non-strongly ergodic and IDPFT.

All necessary definitions can be found in §2.

Remark C. Every IDPFT action is amenable in the sense of Greenleaf. Hence, each Poisson G-action $T^{*}$ that appears in Theorem A(3) or Theorem B(3) is amenable in the sense of Greenleaf. This, in turn, implies that $T^{*}$ is amenable (in the sense of Zimmer) if and only if G is amenable.

We note that Theorem A refines [DaKoRo1, Theorem 8.1], where it was shown that G has property (T) if and only if each non-singular Poisson G-action admits an absolutely continuous invariant probability measure.

We now state a couple of applications of Theorems A and B. First, as far as we know, the following problem is still open.

• • Given a non-amenable G, is there an ergodic (or weakly mixing) non-amenable free non-singular G-action T that does not admit an equivalent invariant probability measure? What is the Krieger type of T?

A partial solution was obtained recently in [ArIsMa], where ergodic non-amenable non-singular G-actions of type $III_{1}$ were constructed for each non-(T) group G. We generalize this result: it follows from Theorem A and Remark C that, for each non-(T) group G, there are weakly mixing non-amenable non-singular Poisson G-actions of each possible Krieger type $K\!\in \mathcal {K}$ .

Second, we introduce the concept of non-singular Bernoulli actions for an arbitrary non-compact locally compact second countable group G. Only the case of discrete G has been considered in the literature so far. By analogy with the probability-preserving case (studied in [OrWe]), we define non-singular Bernoulli actions as non-singular Poisson suspensions of non-singular totally dissipative G-actions. Then, we obtain the following result as a corollary of Theorem B.

Theorem D. Let G be an amenable non-compact locally compact second countable group. If $K\in \{III_{\unicode{x3bb} }\mid 0\le \unicode{x3bb} \le 1\}\cup \{II_{\infty }\}$ , then there is a free non-singular IDPFT Bernoulli G-action of infinite ergodic index and of Krieger type K.

In the case of discrete G, Theorem D was proved in [BeVa, DaKo] (see also earlier works [VaWa] for $K=III_{1}$ and [KoSo] for $K=III_{\unicode{x3bb} }$ with $0<\unicode{x3bb} <1$ ).

Third, we discuss an interplay between non-singular Poisson and non-singular Gaussian actions. While each non-singular G-action generates a unitary Koopman representation of G, each non-singular Poisson G-action T generates also a certain affine Koopman representation of G [DaKoRo1]. The latter gives rise to a non-singular Gaussian G-action S that is unitarily equivalent to T [DaLe, Remark 3.6]. We recall that the non-singular Gaussian G-actions were introduced in [ArIsMa] (see also [DaLe, §3] for an alternative exposition). A natural problem arises: to compare non-spectral dynamical properties of T and S. We show that these properties can be quite different (see also Proposition 9.1 for a slightly more general result).

Proposition E. If G is a non-amenable group with the Haagerup property and $T^{*}$ is a Poisson G-action (of any Krieger type) from Theorem B(3), then the corresponding Gaussian G-action S is weakly mixing, 0-type, amenable in the Greenleaf sense and of Krieger type $III_{1}$ .

Thus, while there are similarities between the theory of non-singular Poisson actions and the theory of non-singular Gaussian actions ^† , Proposition E illustrates that the non-singular Poisson actions have richer orbital structure and they are presumably more suitable for applications in von Neumann algebras ^‡ .

As for the proof of Theorems A and B, we note that, in both cases, the implications $(3)\!\Rightarrow \!(2)\!\Rightarrow \!(1)$ are straightforward. The main result of the paper is to prove that $(1)\Rightarrow (3)$ . Thus, let G be a non-(T) group. We have to construct Poisson G-actions satisfying the properties listed in (3). The idea of the construction is as follows. Let $S=(S_{g})_{g\in G}$ be a measure-preserving G-action on an infinite $\sigma $ -finite measure space $(Y,\kappa )$ . Fix a sequence $(F_{n})_{n=1}^{\infty }$ of subsets of Y such that $\kappa (F_{n})=1$ for each $n\in \Bbb N$ and $\kappa (S_{g}F_{n}\triangle F_{n})\to 0$ uniformly on the compacts in G. The existence of $(Y,\kappa ,S,(F_{n})_{n=1}^{\infty })$ was proved in [Jo] (see also [Da]). Let $(X,\nu , T)$ denote the infinite sum $\bigsqcup _{n=1}^{\infty }(Y,\nu _{n},S)$ of countably many copies of $(Y,\kappa , S)$ , where $\nu _{n}=\kappa $ for each $n\in \Bbb N$ . The Poisson suspension $(X^{*},\nu ^{*}, T^{*})$ of $(X,\nu , T)$ is canonically isomorphic to the infinite direct product $\bigotimes _{n=1}^{\infty }(Y^{*},\nu _{n}^{*}, S^{*})$ of Poisson suspensions of $(Y,\nu _{n}, S)$ . We note that $T^{*}$ preserves $\nu ^{*}$ . Now, we replace $\nu _{n}$ with an equivalent measure $\mu _{n}$ in such a way that:

1. • $\nu _{n}\restriction (Y\setminus F_{n})=\mu _{n}\restriction (Y\setminus F_{n})$ and $\nu _{n}^{*}\sim \mu _{n}^{*}$ for each n;

2. • the product measure $\mu ^{*}:=\bigotimes _{n\in \Bbb N}\mu _{n}^{*}$ is $T^{*}$ -non-singular; and

3. • $\mu ^{*}$ is mutually singular with respect to $\bigotimes _{n\in \Bbb N}\nu _{n}^{*}$ .

All the non-singular Poisson actions from Theorems A(3) and B(3) appear as $(X^{*},\mu ^{*}, T^{*})$ for appropriately chosen ‘parameters’ $(Y,\kappa ,S, (F_{n})_{n=1}^{\infty }, (\mu _{n})_{n=1}^{\infty })$ . Hence, all these Poisson actions have the IDPFT structure: $\mu ^{*}$ splits into an infinite direct product of measures, each of which admits an equivalent invariant probability. We have to show that $T^{*}$ is conservative and ergodic and compute the Krieger type of $T^{*}$ . The other properties are more or less straightforward. First, the conservativeness of $T^{*}$ is established via a (modified) criterion from [DaKoRo1]. Then, we note that, under certain conditions, each conservative IDPFT action is ergodic [DaLe]. In the general case considered in Theorem A, these conditions are not satisfied. However, they are satisfied partly ‘along a direction’. Therefore, we prove ‘directional’ counterparts of the criterion from [DaLe] as auxiliary results. As far as we know, this ‘directional’ approach originates from [ArIsMa], where it was used to show weak mixing of certain non-singular Gaussian actions. Finally, to compute the Krieger type of $T^{*}$ , we use the ‘directional’ approach again, the IDPFT property and the following additional tools:

• • restricted infinite products of probability-preserving systems [Hi] and approximation techniques from [Da] (in the case of type $II_{\infty }$ );

• • Moore–Hill products [Hi] (in the case of type $III_{0}$ ); and

• • asymptotic properties of Skellam distributions (in the case of type $III_{\unicode{x3bb} }$ , $0<\unicode{x3bb} \le 1$ ).

The outline of the paper is as follows. Section 2 contains preliminary results on unitary representations, mixing properties and strong ergodicity of non-singular actions, Krieger type, Maharam extensions and associated flows, Moore–Hill restricted infinite products of probability measures, IDPFT actions, non-singular Poisson suspensions and amenable actions. Some of these results are well known, but some are new. We provide complete proof in the latter case.

In §3, we present a general construction of the Poisson suspension $(X^{*},\mu ^{*},T^{*})$ depending on the parameters $(Y,\kappa ,S, (F_{n})_{n=1}^{\infty }, (\mu _{n})_{n=1}^{\infty })$ . Some conditions on the parameters are found under which $(X^{*},\mu ^{*},T^{*})$ is well defined, non-strongly ergodic, etc.

In §§4, 5, 6 and 7, we prove Theorems A and B in the case $K=II_{\infty }$ , $K=III_{0}$ , $K=III_{\unicode{x3bb} }$ with $0<\unicode{x3bb} <1$ and $K=III_{1}$ , respectively.

In §8, we use Poisson suspensions to introduce non-singular Bernoulli G-actions for arbitrary locally compact second countable group G. When G is discrete, these Bernoulli G-actions are non-singular Bernoulli in the usual sense. Theorem D is proved there.

In the final §9, we prove Proposition E.

2 Definitions and preliminaries

2.1 Unitary representations of G and Koopman representations of non-singular actions

Let $V=(V(g))_{g\in G}$ be a weakly continuous unitary representation of G in a separable Hilbert space $\mathcal {H}$ . We will always assume that V is a complexification of an orthogonal representation of G in a real Hilbert space. We recall several classical concepts from the theory of unitary representations.

Definition 2.1

1. (i) V is called weakly mixing if V has no non-trivial finite-dimensional invariant subspaces.

2. (ii) V is called mixing if $V(g)\to 0$ as $g\to \infty $ in the weak operator topology.

The Fock space $\mathcal {F}(\mathcal {H})$ over $\mathcal {H}$ is the orthogonal sum $\bigoplus _{n=0}^{\infty }\mathcal {H}^{\odot n}$ , where $\mathcal {H}^{\odot n}$ is the n-th symmetric tensor power of $\mathcal {H}$ when $n>0$ and $\mathcal {H}^{\odot 0}:=\Bbb C$ . By $\exp V=(\exp V(g))_{g\in G}$ , we denote the corresponding unitary representation of G in $\mathcal {F}(\mathcal {H})$ , that is, $\exp V(g):=\bigoplus _{n=0}^{\infty } V(g)^{\odot n}$ for each $g\in G$ .

Fact 2.2. The following are equivalent.

1. (i) V is weakly mixing.

2. (ii) There is a sequence $g_{n}\to \infty $ in G such that $V(g_{n})\to 0$ weakly as $n\to \infty $ .

3. (iii) $\exp V\restriction ( \mathcal {F}(\mathcal {H})\ominus \Bbb C)$ is weakly mixing.

The equivalence (i) $\Leftrightarrow $ (ii) follows from [BeRo, Corollary 1.6 and Theorem 1.9]. This equivalence implies immediately the equivalence (i) $\Leftrightarrow $ (iii) (see also [GlWe1, Theorem A3]).

2.2 Non-singular and measure-preserving G-actions

Let $S\kern-2pt =\kern-2pt (S_{g})_{g\in G}$ be a non-singular G-action on a $\sigma $ -finite standard measure space $(Z,\mathfrak {Z},\nu )$ . Denote by $U_{S}=(U_{S}(g))_{g\in G}$ the associated (weakly continuous) unitary Koopman representation of G in $L^{2}(Z,\nu )$ , that is,

$$ \begin{align*} U_{S}(g)f:=f\circ S_{g}^{-1}\sqrt{\frac{d\nu\circ S_{g}^{-1}}{d\nu}}\quad\text{for all }g\in G. \end{align*} $$

All actions in this paper are assumed to be effective (or faithful), that is, $S_{g}=\text {Id}$ if and only if $g=1_{G}$ .

We recall several basic concepts related to non-singular actions of G.

Definition 2.3. Let H be a non-relatively compact subset of G.

1. (i) S is called totally dissipative if the partition of Z into the S-orbits is measurable and the S-stabilizer of almost every (a.e.) point is compact, that is, there is a measurable subset of Z which meets a.e. S-orbit exactly once, and, for a.e. $z\in Z$ , the subgroup $\{g\in G\mid S_{g}z=z\}$ is compact in G.

2. (ii) S is called conservative if there is no any S-invariant subset $A\subset Z$ of positive measure such that the restriction of S to A is totally dissipative.

3. (iii) S is called conservative along H if, for each subset $A\in \mathfrak {Z}$ with $\nu (A)>0$ and each compact subset $K\subset G$ , there is $h\in H\setminus K$ such that $\nu (S_{h}A\cap A)>0$ .

4. (iv) S is called ergodic if each measurable S-invariant subset of Z is either $\mu $ -null or $\mu $ -conull.

5. (v) S is called weakly mixing if, for each ergodic probability-preserving G-action $R=(R_{g})_{g\in G}$ , the product G-action $(S_{g}\times R_{g})_{g\in G}$ is ergodic.

6. (vi) S is of infinite ergodic index if, for each $l>0$ , the the l-th power $(S_{g}^{\times l})_{g\in G}$ of S is ergodic.

7. (vii) S is called of 0-type along H if $U_{S}(g)\to 0$ as $H\ni g\to \infty $ in the weak operator topology.

8. (viii) S is called of 0-type if S is of 0-type along G.

9. (ix) Let S preserve $\nu $ . A sequence $(B_{n})_{n=1}^{\infty }$ of measurable subsets of Z is called S-Følner if, for each compact subset $K\subset G$ ,

   $$ \begin{align*} \lim_{n\to\infty}\sup_{g\in K}\frac{\nu(S_{g}B_{n}\triangle B_{n})}{\nu(B_{n})}=0. \end{align*} $$

10. (x) Let $\nu (Z)=1$ . A sequence $(B_{n})_{n=1}^{\infty }$ of measurable subsets of Z is called S-asymptotically invariant if, for each compact subset $K\subset G$ ,

    $$ \begin{align*} \lim_{n\to\infty} \sup_{g\in K}\nu(B_{n}\triangle S_{g}B_{n})\to 0. \end{align*} $$

11. (xi) Let $\nu (Z)=1$ . Then, S is called strongly ergodic if each S-asymptotically invariant sequence $(B_{n})_{n=1}^{\infty }$ is trivial, that is, $\lim _{n\to \infty }\nu (B_{n})(1-\nu (B_{n}))=0$ .

12. (xii) Let $\nu (Z)=1$ and S preserve $\nu $ . Then, S is called mixing along H if $\lim _{H\ni g\to \infty }\nu (S_{g}A\cap B)=\nu (A)\nu (B)$ for all $A,B\in \mathfrak {Z}$ .

Let $ L^{2}_{0}(Z,\nu ):= L^{2}(Z,\nu )\ominus \Bbb C=\{f\in L^{2}(Z,\nu )\mid \int _{Z}f\,d\nu =0\}$ .

Fact 2.4

1. (i) If the G-action $(S_{g}\times S_{g})_{g\in G}$ is ergodic, then S is weakly mixing [GlWe2, Theorem 1.1].

2. (ii) Let $\nu (Z)=1$ and S preserve $\nu $ . Then, S is weakly mixing if and only if $U_{S}\restriction L^{2}_{0}(Z,\nu )$ is weakly mixing [BeRo].

3. (iii) Given a unitary representation V that is the complexification of an orthogonal representation of G, let $(Y,\mathfrak {C},\nu , T)$ denote the probability-preserving Gaussian dynamical system (G-action) associated with V. Then, there is a canonical unitary equivalence of $U_{T}$ and $\exp V$ [Gu].

We say that a sequence $(g_{k})_{k=1}^{\infty }$ in G is dispersed if, for each compact $K\subset G$ , there is $N>0$ such that if $l>m>N$ , then $g_{l}g_{m}^{-1}\not \in K$ . The following statement is a part of [ArIsMa, Lemma 7.15]. We give an alternative proof of it.

Proposition 2.5. Let S be a non-singular G-action on a probability space $(Z,\mathfrak {Z},\nu )$ . If $(g_{k})_{k=1}^{\infty }$ is a dispersed sequence of G-elements such that $\sum _{k=1}^{\infty }(({d\nu \circ S_{g_{k}}^{-1}})/{d\nu })(z)=+\infty $ at a.e. $z\in Z$ , then S is conservative along the subset $H\kern-0.3pt:=\kern-0.3pt\{g_{l}g_{m}^{-1}\mid l,m\in \Bbb N, l\kern-0.3pt>\kern-0.3ptm\}\kern-0.3pt\subset\kern-0.3pt G$ .

Proof. We first note that, since $(g_{k})_{k=1}^{\infty }$ is dispersed, H is not relatively compact. Let $A\in \mathfrak {Z}$ with $\nu (A)>0$ . Then,

$$ \begin{align*} +\infty= \int_{A}\sum_{k=1}^{\infty}\frac{d\nu\circ S_{g_{k}}^{-1}}{d\nu}(z)\,d\nu(z)=\sum_{k=1}^{\infty} \nu(S_{g_{k}}^{-1}A). \end{align*} $$

Hence, for each $N>0$ , there exist integers $l>m>N$ such that $\nu (S_{g_{l}}^{-1}A\cap S_{g_{m}}^{-1}A)>0$ . Therefore, $\nu (A\cap S_{g_{l}g_{m}^{-1}}A)>0$ . Since $(g_{k})_{k=1}^{\infty }$ is dispersed, it follows that, for each compact subset $K\subset G$ , there is an element $g\in H\setminus K$ such that $\nu (A\cap S_{g}A)>0$ . Hence, S is conservative along H.

The following proposition can be interpreted as a ‘directional’ refinement of the Schmidt–Walters theorem [ScWa]. It was proved in [ArIsMa, Theorem 7.14]. For completeness of our argument, we state the proposition here with a (modified) proof.

Proposition 2.6. Let H be a non-relatively compact subset in G. Let R be a measure-preserving G-action on a standard probability space $(X,\mathfrak {B},\mu )$ and let S be a non-singular G-action on a standard probability space $(Z,\mathfrak {Z},\nu )$ . If R is mixing along H and S is conservative along H, then each $(R_{h}\times S_{h})_{h\in H}$ -invariant subset of $X\times Z$ is the Cartesian product of X with an $(R_{h})_{h\in H}$ -invariant subset of Z.

Proof. Let a subset $A\subset X\times Z$ be $(R_{h}\times S_{h})_{h\in H}$ -invariant and $\mu \otimes \nu (A)\ge 0.5$ . For each $z\in Z$ , we let $A_{z}:=\{x\in X\mid (x,z)\in A\}$ and set $Z_{+}:=\{z\in Z\mid \mu (A_{z})\ge 0.5\}$ . Then, $ \kappa (Z_{+})>0. $ For each $\epsilon>0$ , there exist a countable family $(B_{m})_{m=1}^{\infty }$ of subsets $B_{m}\subset Z$ and a countable family $(C_{m})_{m=1}^{\infty }$ of subsets $C_{m}\subset X$ such that $\nu (B_{m})>0$ , $\bigsqcup _{m=1}^{\infty } B_{m}=Z_{+}$ and $\mu (C_{m}\triangle A_{z})<\epsilon $ for each $z\in B_{m}$ and all $m\in \Bbb N$ . We note that A is $(R_{h}\times S_{h})_{h\in H}$ -invariant if and only if $A_{S_{h}^{-1}z}=R_{h}A_{z}$ for all $h\in H$ at a.e. $z\in Z$ . Fix $m>0$ . Select a sequence $(h_{n})_{n=1}^{\infty }$ in H such that $h_{n}\to \infty $ in G as $n\to \infty $ , $\nu (S_{h_{n}}B_{m}\cap B_{m})>0$ . Take $z_{n}\in S_{h_{n}}B_{m}\cap B_{m}$ , $n\in \Bbb N$ . Then,

$$ \begin{align*} \mu(R_{h_{n}}C_{m}\cap C_{m})=\mu(R_{h_{n}}A_{z_{n}}\cap A_{S_{h_{n}}^{-1}z_{n}})\pm 2\epsilon=\mu(A_{z_{n}})\pm 2\epsilon=\mu(C_{m})\pm3\epsilon. \end{align*} $$

Passing to the limit as $n\to \infty $ , we obtain that

$$ \begin{align*} 3\epsilon\ge |\mu(C_{m})^{2}-\mu(C_{m})|\ge|\mu(C_{m})-1|(0.5-\epsilon). \end{align*} $$

Hence, $\mu (A_{z})>1-12\epsilon $ for all $z\in B_{m}$ . Since m and $\epsilon $ are arbitrary, we conclude that $\mu (A_{z})=1$ for a.e. $z\in Z_{+}$ . Now, consider $(X\times Z)\setminus A$ instead of A. Then, the similar argument yields that $\nu (A_{z})=0$ for a.e. $z\not \in Z_{+}$ . Hence, $A=X\times Z_{+}$ mod 0. Of course, $Z_{+}$ is $(R_{h})_{h\in H}$ -invariant.

2.3 Krieger type

Let $T=(T_{g})_{g\in G}$ be an ergodic non-singular G-action on a standard non-atomic measure space $(X,\mathfrak {B},\mu )$ . The full group $[T,\mu ]$ consists of those $\mu $ -non-singular transformations Q of X for which there is a countable partition $\mathcal {P}$ of X and a map $\mathcal {P}\ni P\mapsto g_{P}\in G$ such that $Qx=T_{g_{P}}x$ at a.e. $x\in P$ for each $P\in \mathcal {P}$ .

If there is a $\mu $ -equivalent $\sigma $ -finite T-invariant measure, then T is called of type $II$ . If the T-invariant measure is finite, then T is called of type $II_{1}$ ; if the T-invariant measure is infinite, then T is called of type $II_{\infty }$ . If T is not of type $II$ , then it is called of type $III$ . The type $III$ admits further classification into subtypes.

We first recall that the Radon–Nikodym cocycle $\rho _{\mu }$ of T is a measurable mapping

$$ \begin{align*} \rho_{\mu}:G\times X\ni (g,x)\mapsto\rho_{\mu}(g,x):=\frac{d\mu\circ T_{g}}{d\mu}(x)\in\Bbb R_{+}^{*}. \end{align*} $$

An element r of the multiplicative group $\Bbb R^{*}_{+}$ is called an essential value of $\rho _{\mu }$ if, for each neighborhood U of r and each subset $A\in \mathfrak {B}$ of positive measure, there exist a subset $B\in \mathfrak {B}$ of positive measure and an element $g\in G$ such that $B\cup T_{g} B\subset A$ and $(({d\mu \circ T_{g}})/{d\mu })(x)\in U$ for each $x\in B$ . The set of all essential values of $\rho _{\mu }$ is denoted by $r(T,\mu )$ . It is a closed subgroup of $\Bbb R_{+}^{*}$ . It is easy to verify that if a measure $\gamma $ is equivalent to $\mu $ , then $r(T,\mu )=r(T,\gamma )$ .

If $r(T,\mu )=\Bbb R_{+}^{*}$ , then T is called of type $III_{1}$ ; if there is $\unicode{x3bb} \in (0,1)$ such that $r(T,\mu )=\{\unicode{x3bb} ^{n}\mid n\in \Bbb Z\}$ , then T is called of type $III_{\unicode{x3bb} }$ . If T is of type $III$ but not of type $III_{\unicode{x3bb} }$ for any $\unicode{x3bb} \in (0,1]$ , then T is called of type $III_{0}$ .

We will need the following folklore approximation result.

Fact 2.7. Let $\mathfrak {B}_{0}\subset \mathfrak {B}$ be a dense subring. Let $\delta>0$ and $s\in \Bbb R_{+}^{*}$ . If, for each $A\in \mathfrak {B}_{0}$ of positive measure and every neighborhood U of s, there is a subset $B\in \mathfrak {B}$ and an element $\theta \in [T,\mu ]$ such that $B\cup \theta B\subset A$ , $\mu (B)>\delta \mu (A)$ and $(({d\mu \circ \theta })/{d\mu })(x)\in U$ for each $x\in B$ , then $s\in r(T,\mu )$ .

2.4 Maharam extension and the associated flow

Let S be a non-singular G-action on a standard probability space $(Z,\mathfrak {Z},\nu )$ . Let $\kappa $ denote the absolutely continuous measure on $\Bbb R$ such that $d\kappa (t)=e^{-t}dt$ . Consider the product space $(\widetilde Z,\widetilde \nu ):=(Z\times \Bbb R,\nu \otimes \kappa )$ . Given $g\in G$ and $s\in \Bbb R$ , we define two transformations, $\widetilde S_{g}$ and $\widetilde s$ of $(\widetilde Z,\widetilde \nu )$ , by setting, for each $(z,t)\in \widetilde Z$ ,

$$ \begin{align*} {\widetilde S}_{g}(z,t):=\bigg(S_{g} z, t+\log\frac{d\nu\circ S_{g}}{d\nu}(z)\bigg)\quad\text{and}\quad \widetilde s(z,t):=(z,t-s). \end{align*} $$

Then, $\widetilde S:=(S_{g})_{g\in G}$ is a measure-preserving G-action. It is called the Maharam extension of T. We note that $(\widetilde s)_{s\in \Bbb R}$ is a totally dissipative non-singular $\Bbb R$ -action on $(\widetilde Z,\widetilde \nu )$ and $\widetilde S_{g}\widetilde s=\widetilde s\widetilde S_{g}$ for all $g\in G$ and $s\in \Bbb R$ . Restrict $(\widetilde s)_{s\in \Bbb R}$ to the $\sigma $ -algebra of $\widetilde S$ -invariant subsets of $Z\times \Bbb R$ and equip this $\sigma $ -algebra with $\widetilde \nu $ (or, more rigorously, with a finite measure which is equivalent to $\widetilde \nu )$ . This restriction is well defined as a non-singular $\Bbb R$ -action. It is called the associated flow of S and is denoted by $W^{S}$ .

Fact 2.8. If $(Z,\nu , S)$ is ergodic, then $W^{S}$ is ergodic. Moreover:

1. (i) S is of type $II$ if and only if $W^{S}$ is transitive and aperiodic;

2. (ii) S is of type $III_{\unicode{x3bb} }$ with $0<\unicode{x3bb} <1$ if and only if $W^{S}$ is transitive but periodic with period $\log \unicode{x3bb} $ ;

3. (iii) S is of type $III_{1}$ if and only if $W^{S}$ is the trivial action on a singleton; and

4. (iv) S is of type $III_{0}$ if and only if $W^{S}$ is non-transitive.

According to the Maharam theorem, the Maharam extension of each conservative dynamical system is conservative. We will need the following ‘directional’ refinement of the Maharam theorem.

Theorem 2.9. Let $(X,\mathfrak {B},\mu , T)$ be a non-singular dynamical system. Let $(g_{k})_{k\in \Bbb N}$ be a dispersed sequence of G-elements. If

(2.1) $$ \begin{align} \sum_{k=1}^{\infty}\bigg(\frac{d\mu\circ T_{g_{k}}}{d\mu}(x)\bigg)^{1+\alpha}=+\infty\quad\text{at a.e. }\textit{x} \end{align} $$

for some real $\alpha \in (0,1)$ , then the Maharam extension $\widetilde T$ of T is conservative along the subset $\{g_{l}g_{m}^{-1}\mid l,m\in \Bbb N, l>m\}$ .

Proof. Let $\tau $ stand for the absolutely continuous probability measure on $\Bbb R$ such that $d\tau (t)=({\alpha }/ 2)e^{-\alpha |t|}dt$ . Then, $\mu \otimes \tau \sim \mu \otimes \kappa $ . For each $k\in \Bbb N$ and $(x,t)\in X\times \Bbb R$ ,

$$ \begin{align*} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)= \frac{d\mu\circ T_{g_{k}}}{d\mu}(x) e^{-\alpha( |t+\log({(d\mu\circ T_{g_{k}})}/{d\mu})(x)|-|t|)}. \end{align*} $$

We now let $N(x,t):=\{k\in \Bbb N\mid (({d\mu \circ T_{g_{k}}})/{d\mu })(x)\ge e^{-t}\}$ . Then, we obtain that

$$ \begin{align*} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)= \begin{cases} e^{(|t|-t)\alpha} \bigg(\frac{d\mu\circ T_{g_{k}}}{d\mu}(x)\bigg)^{1-\alpha} &\text{if }k\in N(x,t),\\ e^{(|t|+t)\alpha} \bigg(\frac{d\mu\circ T_{g_{k}}}{d\mu}(x)\bigg)^{1+\alpha} &\text{if }k\not\in N(x,t). \end{cases} \end{align*} $$

If the set $N(x,t)$ is infinite, then

$$ \begin{align*} \sum_{k\in \Bbb N} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)\ge \sum_{k\in N(x,t)} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)\ge \sum_{k\in N(x,t)} e^{-t(1-\alpha)}=\infty. \end{align*} $$

If $N(x,t)$ is finite, then (2.1) yields that $ \sum _{k\not \in N(x,t)}(({(d\mu \circ T_{g_{k}})}/{d\mu })(x))^{1+\alpha }=+\infty $ and hence

$$ \begin{align*} \sum_{k\in \Bbb N} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)&\ge \sum_{k\not\in N(x,t)} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)\\ &= e^{(|t|+t)\alpha} \sum_{k\not\in N(x,t)}\bigg(\frac{d\mu\circ T_{g_{k}}}{d\mu}(x)\bigg)^{1+\alpha}=+\infty. \end{align*} $$

Thus, in every case, that is, for a.e. $(x,t)\in X\times \Bbb R$ ,

$$ \begin{align*} \sum_{k\in \Bbb N} \frac{d(\mu\otimes\tau)\circ \widetilde T_{g_{k}}}{d(\mu\otimes\tau)}(x,t)=\infty. \end{align*} $$

It remains to apply Proposition 2.5.

2.5 Restricted infinite products of probability measures

Let $(Y,\mathfrak {C})$ be a standard Borel space and let $(\gamma _{n})_{n=1}^{\infty }$ be a sequence of probability measures on $(Y,\mathfrak {C})$ . Let $\boldsymbol {B}\kern-0.5pt:=\kern-0.5pt(B_{n})_{n=1}^{\infty }$ be a sequence of subsets from $\mathfrak {C}$ such that $\gamma _{n}(B_{n})\kern-0.5pt>\kern-0.5pt0$ . We set $(X,\mathfrak {B})\kern-0.5pt:=\kern-0.5pt(Y,\mathfrak {C})^{\otimes \Bbb N}$ . For each $n\in \Bbb N$ , let $ \boldsymbol {B}^{n}:=Y^{n}\times B_{n+1}\times B_{n+2}\times \cdots \in \mathfrak {B}. $ Then, $ \boldsymbol {B}^{1}\subset \boldsymbol {B}^{2}\subset \cdots $ . Define a measure $\gamma ^{ \boldsymbol {B}}$ on $(X,\mathfrak {B})$ by the following sequence of restrictions (see [Hi] for details): that is,

$$ \begin{align*} \gamma^{ \boldsymbol{B}}\restriction\boldsymbol{B}^{n}:=\frac{\gamma_{1}}{\gamma_{1}(B_{1})}\otimes \cdots\otimes \frac{\gamma_{n}}{\gamma_{n}(B_{n})}\otimes \frac{\gamma_{n+1}\restriction B_{n+1}}{\gamma_{n+1}(B_{n+1})}\otimes \frac{\gamma_{n+2}\restriction B_{n+2}}{\gamma_{n+2}(B_{n+2})}\otimes\cdots{,} \end{align*} $$

$n\in \Bbb N$ . Since these restrictions are compatible, $\gamma ^{ \boldsymbol {B}}$ is well defined. We note that $\gamma ^{ \boldsymbol {B}}$ is supported on the subset $\bigcup _{n=1}^{\infty } \boldsymbol {B}^{n}\subset X$ and $\gamma ^{ \boldsymbol {B}}(\boldsymbol {B}^{n})=\prod _{j=1}^{n}\gamma _{j}(B_{j})^{-1}$ for each n. Hence, $\gamma ^{ \boldsymbol {B}}$ is $\sigma $ -finite. It is infinite if and only if $\prod _{n=1}^{\infty } \gamma _{n}(B_{n})=0$ .

Definition 2.10. [Hi]

1. (i) We call $\gamma ^{ \boldsymbol {B}}$ the restricted infinite product of $(\gamma _{n})_{n=1}^{\infty }$ with respect to $\boldsymbol {B}$ .

2. (ii) A $\sigma $ -finite measure $\kappa $ on $(X,\mathfrak {B})$ is called a Moore–Hill-product of $(\gamma _{n})_{n=1}^{\infty }$ if, for each $n\in \Bbb N$ , there exists a $\sigma $ -finite measure $\kappa _{n}$ on the infinite product space $\bigotimes _{k>n}(Y,\mathfrak {C})$ such that $\gamma =\gamma _{1}\otimes \cdots \otimes \gamma _{n}\otimes \kappa _{n}$ .

Let $T=(T_{g})_{g\in G}$ be a Borel action of G on a Borel space $(Y,\mathfrak {C},\gamma )$ . Let $\gamma _{n}\circ T_{g}=\gamma _{n}$ for each $g\in G$ and each $n\in \Bbb N$ . We define a Borel G-action $\boldsymbol {T}=(\boldsymbol {T}_{g})_{g\in G}$ on $(X,\mathfrak {B}):=(Y,\mathfrak {C})^{\otimes \Bbb N}$ by setting $\boldsymbol {T}_{g}:=\bigotimes _{n=1}^{\infty } T_{g}$ for each $g\in G$ .

Fact 2.11. (See [Da, Proposition 2.6])

If $\sum _{n=1}^{\infty }({\gamma _{n}(B_{n}\triangle T_{g}B_{n})}/{\gamma _{n}(B_{n})})<\infty $ for some $g\in G$ , then $\boldsymbol {T}_{g}$ preserves $\gamma ^{ \boldsymbol {B}} $ .

The following formula is checked straightforwardly for each $g\in G$ and $n\in \Bbb N$ : that is,

(2.2) $$ \begin{align} \frac{\gamma^{\boldsymbol{B}}(\boldsymbol{T}_{g}\boldsymbol{B}^{n}\cap\boldsymbol{B}^{n})}{\gamma^{\boldsymbol{B}}(\boldsymbol{B}^{n})}=\prod_{j>n}\frac{\gamma_{j}(T_{g}B_{j}\cap B_{j})}{\gamma_{j}(B_{j})}. \end{align} $$

Hence, if, for each compact $K\subset G$ ,

$$ \begin{align*} \sum_{n=1}^{\infty}\sup_{g\in K}\frac{\gamma_{n}(B_{n}\triangle T_{g}B_{n})}{\gamma_{n}(B_{n})}<\infty, \end{align*} $$

then the sequence $(\boldsymbol {B}^{n})_{n=1}^{\infty }$ is $\boldsymbol {T}$ -Følner.

Let $U_{j,T}$ and $U_{\boldsymbol {T}}$ denote the unitary Koopman representations of G in $L^{2}(Y,\gamma _{j})$ and $L^{2}(X,\gamma ^{\boldsymbol {B}})$ associated with T and $\boldsymbol {T}$ , respectively.

Fact 2.12. (Cf. [Da, Lemma 2.7])

Let $\sum _{n=1}^{\infty }({\gamma _{n}(B_{n}\triangle T_{g}B_{n})}/{\gamma _{n}(B_{n})})<\infty $ . Then, for each $n\in \Bbb N$ and two arbitrary functions $f,q\in L^{2}(Y^{n},\bigotimes _{j=1}^{n}\gamma _{j})$ ,

$$ \begin{align*} \bigg\langle U_{\boldsymbol{T}}(g)\bigg(f\otimes \bigotimes_{j>n}1_{B_{j}}\bigg),q\otimes \bigotimes_{j>n}1_{B_{j}}\bigg\rangle =\frac{\langle(\bigotimes_{j=1}^{n}U_{j,T})(g)f,q\rangle}{\prod_{j=1}^{n}\gamma_{j}(B_{j})}\prod_{j>n}\frac{\gamma_{j}(T_{g}B_{j}\cap B_{j})}{\gamma_{j}(B_{j})}. \end{align*} $$

Corollary 2.13. Let $\sum _{n=1}^{\infty }({\gamma _{n}(B_{n}\triangle T_{g}B_{n})}/{\gamma _{n}(B_{n})})<\infty $ and $\prod _{n=1}^{\infty }\gamma _{n}(B_{n})=0$ (that is, $\gamma ^{\boldsymbol {B}}$ is infinite). If $(Y,\gamma _{n},T)$ is mixing along a non-relatively compact subset $H\subset G$ for each $n\in \Bbb N$ , then $\boldsymbol {T}$ is of 0-type along H.

Proof. Given two positive integers $n<m$ ,

$$ \begin{align*} \lim_{H\ni g\to\infty}\prod_{j=n+1}^{m}\frac{\gamma_{j}(T_{g}B_{j}\cap B_{j})}{\gamma_{j}(B_{j})}=\prod_{j=n+1}^{m}\gamma_{j}(B_{j}) \end{align*} $$

because T is mixing along H. Hence, $\lim _{H\ni g\to \infty }\prod _{j>n}{\gamma _{j}(T_{g}B_{j}\cap B_{j})}/{\gamma _{j}(B_{j})}=0$ because $\gamma ^{\boldsymbol {B}}$ is infinite. Therefore, it follows from Fact 2.12 that $U_{\boldsymbol {T}}(g)\to 0$ weakly as $H\ni g\to \infty $ .

Given two probability measures $\alpha ,\beta $ on a standard Borel space $(Y,\mathfrak {C})$ , let $\gamma $ be a third probability measure on $\mathfrak {C}$ such that $\alpha \prec \gamma $ and $\beta \prec \gamma $ . The (squared) Hellinger distance between $\alpha $ and $\beta $ is

$$ \begin{align*} H^{2}(\alpha,\beta):=\frac12\int_{Y}\bigg(\sqrt{\frac{d\alpha}{d\gamma}}-\sqrt{\frac{d\beta}{d\gamma}}\bigg)^{2}=1-\int_{Y}\sqrt{\frac{d\alpha}{d\gamma}\frac{d\beta}{d\gamma}}\,d\gamma. \end{align*} $$

This definition does not depend on the choice of $\gamma $ . The Hellinger distance is used in the Kakutani theorem on equivalence of infinite products of probability measures [Ka]. We will utilize the following fact, which is an extension of the Kakutani theorem.

Fact 2.14. Let $(\gamma _{n})_{n=1}^{\infty }$ and $(\alpha _{n})_{n=1}^{\infty }$ be two sequences of probability measures on $(Y,\mathfrak {C})$ . Let $\boldsymbol {B}=(B_{n})_{n=1}^{\infty }$ be a sequence of subsets $B_{n}\in \mathfrak {C}$ such that $\gamma _{n}(B_{n})>0$ . Then, the following are satisfied.

1. (i) [Hi, Theorem 3.6] $\gamma ^{\boldsymbol {B}}\sim \bigotimes _{n=1}^{\infty }\alpha _{n}$ if and only if $\gamma _{n}\sim \alpha _{n}$ for each $n\in \Bbb N$ and

   $$ \begin{align*}\sum_{n=1}^{\infty} H^{2}\bigg(\frac 1{\gamma_{n}(B_{n})}(\gamma_{n}\restriction B_{n}),\alpha_{n}\bigg)<\infty.\end{align*} $$

2. (ii) [Hi, Theorem 3.9] Let $\gamma $ be a Moore–Hill product of $(\gamma _{n})_{n=1}^{\infty }$ . Then, there exist $a>0$ and a sequence $\boldsymbol {D}=(D_{n})_{n=1}^{\infty }$ of subsets $D_{n}\in \mathfrak {C}$ such that $\gamma =a\gamma ^{\boldsymbol {D}}$ .

We also note that if $\gamma ^{\boldsymbol {B}}\sim \bigotimes _{n=1}^{\infty }\alpha _{n}$ , then $(\bigotimes _{n=1}^{\infty }\alpha _{n})(\boldsymbol B^{k})>0$ for each $k>0$ . In particular, $\prod _{n=1}^{\infty }\alpha _{n}(B_{n})>0$ .

2.6 IDPFT actions

IDPFT actions were introduced in [DaLe] in the case where $G=\Bbb Z$ . IDPFT actions of arbitrary discrete countable groups and arbitrary locally compact Polish groups were under consideration in [Da, DaKo], respectively.

Definition 2.15. Let $S_{n}=(S_{n}(g))_{g\in G}$ be an ergodic measure-preserving G-action on a standard probability space $(Z_{n},\mathfrak {Z}_{n},\nu _{n})$ , let $\mu _{n}$ be a probability measure on $\mathfrak {C}_{n}$ and let $\mu _{n}\sim \nu _{n}$ for each $n\in \Bbb N$ . We set $(Z,\mathfrak {Z},\nu ):=\bigotimes _{n=1}^{\infty }(Z_{n},\mathfrak {Z}_{n},\nu _{n})$ , $\mu :=\bigotimes _{n=1}^{\infty }\mu _{n}$ , $S(g):=\bigotimes _{n=1}^{\infty } S_{n}(g)$ for each $g\in G$ and $S:=(S(g))_{g\in G}$ . If $\mu \circ S(g)\sim \mu $ for each $g\in G$ , then the non-singular dynamical system $(Z,\mathfrak {Z},\mu , S)$ is called an infinite direct product of finite types (IDPFT).

The Radon–Nikodym cocycle of an IDPFT system is an infinite product

$$ \begin{align*} \frac{d\mu\circ S(g)}{d\mu}(z)=\prod_{n=1}^{\infty}\frac{d\mu_{n}\circ S_{n}(g)}{d\mu_{n}}(z_{n})\quad\text{at }\mu\text{-a.e. }z=(z_{n})_{n=1}^{\infty}\in Z\text{, }g\in G. \end{align*} $$

We will need the following fact, which partially extends the results [DaLe, Proposition 2.3] and [Da, Proposition 2.9] about sharp weak mixing of conservative IDPFT systems with mildly mixing factors. We extend these results to the IDPFT systems whose factors are mixing along some ‘directions’.

Proposition 2.16. Let $(Z,\mathfrak {Z},\mu , S)$ be an IDPFT system as in Definition 2.15. Let $(g_{k})_{k=1}^{\infty }$ be a dispersed sequence of G-elements such that $\sum _{k=1}^{\infty }(({d\mu \circ S(g_{k})^{-1}})/{d\mu })(z)=+\infty $ at a.e. $z\in Z$ . Let $H:=\{g_{l}g_{m}^{-1}\mid l,m\in \Bbb N, l>m\}$ . If the system $(Z_{n},\mathfrak {Z}_{n},\nu _{n}, S_{n})$ is mixing along H for each $n\in \Bbb N$ , then $(Z,\mathfrak {Z},\mu , S)$ is weakly mixing.

Proof. Let $R=(R_{g})_{g\in G}$ be an ergodic measure-preserving action of G on a standard probability space $(Y,\mathfrak {Y},\kappa )$ . Then,

$$ \begin{align*} \sum_{k=1}^{\infty}\frac{d(\mu\otimes\kappa)\circ (S({g_{k}})^{-1}\times R_{g_{k}}^{-1})}{d(\mu\otimes\kappa)}(z,y)=\sum_{k=1}^{\infty}\frac{d\mu\circ S({g_{k}})^{-1}}{d\mu}(z)=+\infty \end{align*} $$

at a.e. $(z,y)\in Z\times Y$ . It follows from Proposition 2.5 that the product G-action $S\times R:=(S_{g}\times R_{g})_{g\in G}$ on the space $(Z\times Y,\mu \otimes \kappa )$ is conservative along H. Let $A\subset Z\times Y$ be an $(S\times R)$ -invariant subset of positive measure. Fix $n\in \Bbb N$ . Then, the product G-action $((\bigotimes _{j>n}S_{j}(g))\otimes R_{g})_{g\in G}$ on $((\prod _{j>n}Z_{j})\otimes Y,(\bigotimes _{j>n}\mu _{n})\otimes \kappa )$ is also conservative along H. On the other hand, the G-action $(\bigotimes _{j=1}^{n}S_{j}(g))_{g\in G}$ on the space $(\prod _{j=1}^{n}Z_{j},\bigotimes _{j=1}^{n}\nu _{n})$ is mixing along H. Since the probability measure $(\bigotimes _{j=1}^{n}\nu _{n})\otimes (\bigotimes _{j>n}\mu _{n})$ on Z is equivalent to $\mu $ , it follows from Proposition 2.6 that there is a subset $A_{n}\subset (\prod _{j>n}Z_{j})\times Y$ of positive measure such that $A=(\prod _{j=1}^{n}Z_{j})\times A_{n}$ . Since n is arbitrary, it follows that $A=Z\times B$ for some subset $B\in \mathfrak {Y}$ invariant under R. Since R is ergodic, $B=Y$ and hence $A=Z\times Y$ . Hence, $S\times R$ is ergodic, that is, S is weakly mixing.

Let Aut $(Z,\mu )$ denote the group of all invertible $\mu $ -non-singular transformations of Z. We recall that the weak topology on Aut $(Z,\mu )$ is induced by the weak operator topology on the unitary group $\mathcal {U}$ of $L^{2}(Z,\mu )$ via the embedding

$$ \begin{align*} \text{Aut}(Z,\mu)\ni R\mapsto U_{R}\in \mathcal{U}. \end{align*} $$

Then, Aut $(Z,\mu )$ is a Polish group under the weak topology.

Let $\bigoplus _{n=1}^{\infty } G$ stand for the direct sum of countably many copies of G. We endow this group with the topology of inductive limit. Then, $\bigoplus _{n=1}^{\infty } G$ is $\sigma $ -finite but not locally compact. In fact, it is not metrizable. Nevertheless, the non-singular actions of $\bigoplus _{n=1}^{\infty } G$ are well defined. Every such action V is nothing but a collection $(V_{n})_{n=1}^{\infty }$ of countable many mutually commuting G-actions defined on the same measure space. We then write $V=\bigoplus _{n=1}^{\infty } V_{n}$ . Hence, the Maharam extension of V, which we denote by $\widetilde V$ , is also well defined. Therefore, we can construct the associated flow of V in the same way as we do for the actions of locally compact groups.

The following proposition is an analog of [DaLe, Theorem 2.10] and [DaKo, Proposition 1.6] for IDPFT systems with non-mildly mixing factor actions. We provide a proof for completeness of our argument.

Proposition 2.17. Let $(Z,\mathfrak {Z},\mu , S)$ be an IDPFT system as in Definition 2.15. Let $(g_{k})_{k=1}^{\infty }$ be a dispersed sequence of G-elements such that

$$ \begin{align*} \sum_{k=1}^{\infty}\bigg(\frac{d\mu\circ S({g_{k}})^{-1}}{d\mu}(z)\bigg)^{\vartheta}=+\infty\quad\text{at a.e. }z\in Z \end{align*} $$

for some $\vartheta \in (1,2)$ . Let $H:=\{g_{l}g_{m}^{-1}\mid l,m\in \Bbb N, l>m\}$ . If the system $(Z_{n},\mathfrak {Z}_{n},\nu _{n}, S_{n})$ is mixing along H for each $n\in \Bbb N$ , then the $\sigma $ -algebra $\mathcal {I}(\widetilde S)$ of $\widetilde S$ -invariant subsets equals the $\sigma $ -algebra $\mathcal {I}(\widetilde {\bigoplus _{n=1}^{\infty } S_{n}})$ of $\widetilde {\bigoplus _{n=1}^{\infty } S_{n}}$ -invariant subsets. Hence, the associated flow of S coincides with the associated flow of $\bigoplus _{n=1}^{\infty } S_{n}$ .

Proof. We first note that it follows from the condition of the proposition and Theorem 2.9 that the Maharam extension $\widetilde S$ of S is conservative along H.

Take a subset $A\in \mathcal {I}(\widetilde S)$ . For every $n\in \Bbb N$ , we define a measure-preserving map $E_{n}:(Z\times \Bbb R,\mu \otimes \eta )\to (Z\times \Bbb R, (\bigotimes _{j=1}^{n}\nu _{j})\otimes (\bigotimes _{j>n}\mu _{n})\otimes \eta )$ by setting

$$ \begin{align*} E_{n}(z,t):=\bigg(x,t+\sum_{j=1}^{n}\log\frac{d\mu_{j}}{d\nu_{j}}(z)\bigg). \end{align*} $$

It follows that, for each $g\in G$ ,

$$ \begin{align*} E_{n}\widetilde S(g)E_{n}^{-1}=\bigg(\bigotimes_{j=1}^{n} S_{j}(g)\bigg)\otimes \widetilde{\bigotimes_{j>n} S_{j}}(g) \end{align*} $$

and that the subset $E_{n}A$ is invariant under $E_{n}\widetilde S(g)E_{n}^{-1}$ . Since $\widetilde S$ is conservative along H, the action $(E_{n}\widetilde S(g)E_{n}^{-1})_{g\in G}$ is also conservative along H. We note that the G-action $(\widetilde {\bigotimes _{j>n} S_{j}}(g))_{g\in G}$ on the space $(\bigotimes _{j>n}(Z_{n},\mu _{n}))\otimes (\Bbb R,\eta )$ is a quotient (that is, a factor) of $(E_{n}\widetilde S(g)E_{n}^{-1})_{g\in G}$ . Hence, $(\widetilde {\bigotimes _{j>n} S_{j}}(g))_{g\in G}$ is also conservative along H. On the other hand, the measure-preserving G-action $(\bigotimes _{j=1}^{n} S_{j}(g))_{g\in G}$ on the probability space $\bigotimes _{j=1}^{n}(Z_{n},\mathfrak {Z}_{n},\nu _{n})$ is mixing along H. Hence, by Proposition 2.6, $E_{n}A=(\bigotimes _{j=1}^{n} Y_{j})\times A_{n}$ for some subset $A_{n}\subset (\bigotimes _{j>n} Y_{j})\times \Bbb R$ . In particular, $E_{n}A$ is invariant under the $G^{n}$ -action $(\bigoplus _{j=1}^{n} S_{j})\otimes I$ . Hence, A is invariant under $E_{n}^{-1}((\bigoplus _{j=1}^{n} S_{j})\otimes I)E_{n}$ , which is exactly the Maharam extension of the $G^{n}$ -action $(\bigoplus _{j=1}^{n} S_{j})\otimes I$ on $(Z,\mathfrak {Z},\mu )$ . Since n is arbitrary, $A\in {\mathcal {I}}(\widetilde {\bigoplus _{n=1}^{\infty } S_{n}})$ , as desired.

Conversely, let $A\in {\mathcal {I}}(\widetilde {\bigoplus _{n=1}^{\infty } S_{n}})$ . We note that, for each $g\in G$ , the sequence of transformations $(S_{1}(g)\times \cdots \times S_{n}(g)\times I)_{n=1}^{\infty }$ converges weakly to the transformation $S(g)$ in Aut $(Z,\mu )$ . Since the mapping

$$ \begin{align*} \text{Aut}(Z,\mu)\ni Q\mapsto \widetilde Q\in\text{Aut}(Z\times \Bbb R,\mu\otimes\tau) \end{align*} $$

is weakly continuous, the Maharam extension of the transformation ${S(g)}$ is the weak limit of the sequence of Maharam extensions of the transformations $S_{1}(g)\times \cdots \times S_{n}(g)\times I$ as $n\to \infty $ . It follows that $A\in \mathcal {I}(\widetilde S)$ , and the proof is complete.

2.7 Non-singular Poisson suspension

Let $(X,\mathfrak {B})$ be a standard Borel space and let $\mu $ be an infinite $\sigma $ -finite non-atomic measure on X. Let $X^{*}$ be the set of purely atomic ( $\sigma $ -finite) measures on X. For each subset $A\in \mathfrak {B}$ with $0<\mu (A)<\infty $ , we define a mapping $N_{A}:X^{*}\to \Bbb R$ by setting $N_{A}(\omega ):=\omega (A)$ . Let $\mathfrak {B}^{*}$ stand for the smallest $\sigma $ -algebra on $X^{*}$ such that the mappings $N_{A}$ are all $\mathfrak {B}^{*}$ -measurable. There is a unique probability measure $\mu ^{*}$ on $(X^{*},\mathfrak {B}^{*})$ satisfying the following two conditions.

• • The measure $\mu ^{*}\circ N_{A}^{-1}$ is the Poisson distribution with parameter $\mu (A)$ for each $A\in \mathfrak {B}$ with $\infty>\mu (A)>0$ .

• • Given a finite family $A_{1},\ldots , A_{q}$ of mutually disjoint subsets $A_{1},\ldots , A_{q}\in \mathfrak {B}$ of finite positive measure, the corresponding random variables $N_{A_{1}},\ldots , N_{A_{q}}$ defined on the space $(X^{*},\mathfrak {B}^{*}, \mu ^{*})$ are independent.

Then, $(X^{*},\mathfrak {B}^{*}, \mu ^{*})$ is a Lebesgue space. Given a subset $B\in \mathfrak {B}$ and an integer $n\in \Bbb Z_{+}$ , we denote by $[B]_{n}$ the cylinder $\{\omega \in X^{*}\mid \omega (B)=n\}$ . We now let

$$ \begin{align*} \text{Aut}_{1}(X,\mu):=\bigg\{S\in \text{Aut}(X,\mu)\,\bigg|\, {\frac{d\mu\circ S}{d\mu}}-1\in L^{1}(X,\mu)\bigg\}. \end{align*} $$

If $S\in \text {Aut}_{1}(X,\mu )$ , we put $\chi (S):=\int _{X}( {(({d\mu \circ S})/{d\mu })}-1)d\mu $ . Then, Aut $_{1}(X,\mu )$ is a subgroup of Aut $(X,\mu )$ and $\chi $ is a homomorphism of Aut $_{1}(X,\mu )$ onto $\Bbb R$ . Suppose that $T=(T_{g})_{g\in G}$ is a non-singular G-action on $(X,\mathfrak {B},\mu )$ . We now define a Borel transformation $T_{g}^{*}$ of $X^{*}$ by setting

$$ \begin{align*} T_{g}^{*}\omega:=\omega\circ T_{g}^{-1} \quad\text{for all }\omega\in X^{*}\text{ for each }g\in G\text{.} \end{align*} $$

Fact 2.18

1. (i) If $T_{g}\in \textrm {{Aut}}_{1}(X,\mu )$ for each $g\in G$ , then $T^{*}:=(T_{g}^{*})_{g\in G}$ is a well defined non-singular G-action on $(X^{*},\mathfrak {B}^{*}, \mu ^{*})$ [DaKoRo1, §§6.1 and 4].

2. (ii) If $\nu $ is a $\sigma $ -finite measure on $(X,\mathfrak {B})$ , then $\nu ^{*}\sim \mu ^{*}$ if and only if $\mu \sim \nu $ and $\sqrt {{d\mu }/{d\nu }}-1\in L^{2}(X,\nu )$ (see [Ta] or [DaKoRo1, Theorem 3.3]).

3. (iii) Moreover, if ${d\mu }/{d\nu }-1\in L^{1}(X,\nu )$ , then

   $$ \begin{align*} \frac{d\mu^{*}}{d\nu^{*}}(\omega)=e^{-\int_{X}({d\mu}/{d\nu}-1)\,d\nu}\prod_{\omega(\{x\})=1}\frac{d\mu}{d\nu}(x)\quad\text{at a.e. }\omega\in X^{*}\text{.} \end{align*} $$

4. (iv) It follows (see also [DaKoRo1, Corollary 4.1(3)]) that if $S\in \textrm {{Aut}}_{1}(X,\mu )$ and $\chi (S)=0$ , then

   $$ \begin{align*} \frac{d\mu^{*}\circ (S^{*})^{-1}}{d\mu^{*}}(\omega)=\prod_{\{x\in X\mid\omega(\{x\})=1\}}\frac{d\mu\circ S^{-1}}{d\mu}(x)\quad\text{for a.e. }\omega\in X^{*}. \end{align*} $$

5. (v) If X is partitioned into countably many T-invariant subsets $X_{n}$ , $n\in \Bbb N$ , then $(X^{*},\mu ^{*},T^{*})$ is canonically isomorphic to the direct product $\bigotimes _{n=1}^{\infty }(X_{n}^{*},\mu _{n}^{*},T_{n}^{*})$ , where $\mu _{n}$ and $T_{n}$ denote the restriction of $\mu $ and T, respectively, to $X_{n}$ .

In particular, if T preserves $\mu $ , then $T^{*}$ preserves $\mu ^{*}$ .

Definition 2.19. The dynamical system $(X^{*},\mathfrak {B}^{*}, \mu ^{*}, T^{*})$ is called the non-singular Poisson suspension of $(X,\mathfrak {B}, \mu , T)$ ^† . A non-singular G-action is called Poisson if it is isomorphic to the Poisson suspension of some non-singular G-action (see [DaKoRo1] for details).

The following proposition is an adaptation of [DaKo, Lemma 1.3] (see also [DaKoRo2, Theorem 3.4]) to the case of locally compact group actions.

Proposition 2.20. Let $T=(T_{g})_{g\in G}$ be a non-singular G-action on a $\sigma $ -finite measure standard non-atomic measure space $(X,\mathfrak {B},\mu )$ . Suppose that $T_{g}\in \textrm {{Aut}}_{1}(X,\mu )$ and $\chi (T_{g})=0$ and for all $g\in G$ . Let $(g_{k})_{k\in \Bbb N}$ be a dispersed sequence of G-elements such that $({d\mu }/({d\mu \circ T_{g_{k}}^{-1}}))^{2}-1\in L^{1}(X,\mu )$ . If there is a sequence $(b_{k})_{k=1}^{\infty }$ of positive reals such that $b_{k}\le 1$ for each k, $\sum _{k=1}^{\infty } b_{k}=+\infty $ but

(2.3) $$ \begin{align} \sum_{k=1}^{\infty} b_{k}^{1+\vartheta}e^{\int_{X}(({d\mu}/({d\mu\circ T_{g_{k}}^{-1}}))^{2}-1)\,d\mu}<+\infty \end{align} $$

for some real $\vartheta>0$ , then

(2.4) $$ \begin{align} \sum_{k=1}^{\infty}\bigg(\frac{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}{d\mu^{*}}(\omega)\bigg)^{2/({1+\vartheta})}=+\infty\quad\text{at a.e. }\omega\text{.} \end{align} $$

If $\vartheta \le 1$ , then the Poisson suspension $T^{*}:=(T_{g}^{*})_{g\in G}$ of T is conservative along the subset $\{g_{l}g_{m}^{-1}\mid l,m\in \Bbb N, l>m\}$ .

Proof. As in the the proof of [DaKo, Lemma 1.3], it follows from the assumptions of the lemma that, for each $k\in \Bbb N$ ,

$$ \begin{align*} M_{k} := \bigg\|\frac{d\mu^{*}}{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}\bigg\|_{2}^{2} = e^{\int_{X}(({d\mu}/({d\mu\circ T_{g_{k}}^{-1}}))^{2}-1)\,d\mu}. \end{align*} $$

By Markov’s inequality,

$$ \begin{align*} \mu^{*}\bigg(\bigg\{\omega\in X^{*}\,\bigg|\, \frac{d\mu^{*}}{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}(\omega)>b_{k}^{-( {1+\vartheta})/2} \bigg\}\bigg) &\le b_{k}^{1+\vartheta} M_{k}\\ &=b_{k}^{1+\vartheta} e^{\int_{X}(( {d\mu}/({d\mu\circ T_{g_{k}}^{-1}}))^{2}-1)\,d\mu}. \end{align*} $$

It now follows from (2.3) and the Borel–Cantelli lemma that

(2.5) $$ \begin{align} \frac{d\mu^{*}}{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}(\omega)\le b_{k}^{-({1+\vartheta})/2} \end{align} $$

for all but finitely many $k\in \Bbb N$ at a.e. $\omega $ . We can rewrite (2.5) as

$$ \begin{align*} \bigg(\frac{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}{d\mu^{*}}(\omega)\bigg)^{{2}/({1+\vartheta})}\ge b_{k}. \end{align*} $$

As $\sum _{k=1}^{\infty } b_{k}=\infty $ , (2.4) follows.

If $\vartheta \le 1$ , then $({1+\vartheta })/2\le 1$ and we deduce from (2.4) that

$$ \begin{align*} \sum_{k=1}^{\infty}\frac{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}{d\mu^{*}}(\omega) \ge \sum_{k=1}^{\infty} b_{k}^{({1+\vartheta})/2}\ge \sum_{k=1}^{\infty} b_{k} =+\infty \end{align*} $$

at a.e. $\omega $ . It remains to apply Proposition 2.5.

Denote by $U_{T}$ and $U_{T^{*}}$ the corresponding unitary Koopman representations of G in $L^{2}(X,\mu )$ and $L^{2}(X^{*},\mu ^{*})$ , respectively.

Fact 2.21. (See [Ro])

There is a canonical unitary isomorphism of $U_{T^{*}}$ and $\exp U_{T}$ .

2.8 Amenability of groups and actions

Let $T=(T_{g})_{g\in G}$ be a non-singular G-action on standard probability space $(X,\mathfrak {B},\mu )$ . Denote by $\unicode{x3bb} _{G}$ a left Haar measure on G.

Definition 2.22

1. (i) T is called amenable [AD, Zi] if there is a G-invariant mean (that is, a positive unital linear map) from $L^{\infty }(X\times G,\mu \otimes \unicode{x3bb} _{G})$ onto $L^{\infty }(X,\mu )$ , where $X\times G$ is endowed with the diagonal G-action (with G acting on G by left rotations).

2. (ii) T is called amenable in the Greenleaf sense [AD, Gr] if there is a G-invariant mean on $L^{\infty }(X,\mu )$ .

Of course, each probability-preserving G-action is amenable in the Greenleaf sense. We also recall the definition of weak containment for unitary representations.

Definition 2.23. Let $\pi _{1}$ and $\pi _{2}$ be two unitary representations of G in Hilbert spaces $\mathcal {H}_{1}$ and $\mathcal {H}_{2}$ , respectively. $\pi _{1}$ is weakly contained in $\pi _{2}$ if, given a vector $h\in \mathcal {H}_{1}$ , a compact subset $K\subset G$ and $\epsilon>0$ , there exist finitely many vectors $h_{1},\ldots , h_{k}\in \mathcal {H}_{2}$ such that

$$ \begin{align*} \sup_{g\in K}\bigg| \langle \pi_{1}(g)h,h\rangle-\sum_{j=1}^{k} \langle \pi_{2}(g)h_{j},h_{j}\rangle\bigg|<\epsilon. \end{align*} $$

Let $L_{G}$ stand for the left regular representation of G.

Fact 2.24. Let T be a non-singular G-action.

1. (i) If G is amenable, then T is amenable [Zi].

2. (ii) If T is amenable, then $U_{T}$ is weakly contained in $L_{G}$ [AD, Theorem 4.3.1].

3. (iii) $U_{T}$ has almost invariant vectors if and only if the trivial representation of G is weakly contained in $U_{T}$ .

4. (iv) G is amenable if and only if the trivial representation of G is weakly contained in $L_{G}$ .

5. (v) T is amenable in the Greenleaf sense if and only if the trivial representation of G is weakly contained in $U_{T}$ [AD, Proposition 4.1.1].

The converse to (i) does not hold. For each locally compact second countable group G, the action of G on itself by left translations is amenable. The converse to (ii) does not hold either [AD].

Proposition 2.25. Let $(Z,\mathfrak {Z},\mu ,S)$ be an IDPFT system as in Definition 2.15. Then, S is amenable in the Greenleaf sense. Hence, S is amenable if and only if G is amenable.

Proof. We will use the notation from Definition 2.15. Let $\xi _{n}:={d\mu _{n}}/{d\nu _{n}}$ for each $n\in \Bbb N$ . We will show that the Koopman representation $U_{S}$ has almost invariant vectors. Fix a compact subset $K\subset G$ and a real $\epsilon>0$ . Since $\min _{g\in K}\langle U_{S}(g)1,1\rangle>0$ and $\langle U_{S}(g)1,1\rangle = \prod _{k=1}^{\infty }\langle U_{S_{k}}(g)1,1\rangle $ , there is $N\in \Bbb N$ such that

$$ \begin{align*} \sup_{g\in K}\bigg|\prod_{k>N}\langle U_{S_{k}}(g)1,1\rangle-1\bigg|<\epsilon. \end{align*} $$

Let $v_{n}:= 1/{ \sqrt {\xi _{n}}}$ . It is straightforward to verify that $v_{n}\in L^{2}(Z_{n},\mu _{n})$ , $\|v_{n}\|_{2}=1$ and $ \langle U_{S_{n}}(g)v_{n},v_{n}\rangle =1 $ for each $g\in G$ . Let $w_{n}:=v_{1}\otimes \cdots \otimes v_{n}\otimes 1\otimes 1\otimes \cdots $ . Then, $w_{n}\in L^{2}(Z,\mu )$ and $\|w_{n}\|_{2}=1$ for each $n\in \Bbb N$ . Moreover,

$$ \begin{align*} \langle U_{S}(g)w_{N},w_{N}\rangle=\prod_{k>N}\langle U_{S_{k}}(g)1,1\rangle=1\pm\epsilon \end{align*} $$

for each $g\in K$ . Hence, $\|U_{S}(g)w_{N}-w_{N}\|^{2}<2\epsilon $ . Thus, $U_{S}$ has almost invariant vectors. In view of Fact 2.24(iii) and (v), S is amenable in the Greenleaf sense.

If G is amenable, then S is amenable by Fact 2.24(i).

Now, suppose that G is non-amenable but S is amenable. Since $U_{S}$ has almost invariant vectors, it follows from Fact 2.24(iii) that the trivial representation of G is weakly contained in $U_{S}$ . Since S is amenable, $U_{S}$ is weakly contained in $L_{G}$ by Fact 2.24(ii). Hence, the trivial representation of G is weakly contained in $L_{G}$ , that is, G is amenable by Fact 2.24(iv), which is a contradiction.

3 General construction

Let $S=(S_{g})_{g\in G}$ be a measure-preserving G-action on an infinite $\sigma $ -finite standard measure space $(Y,\mathfrak {Y},\kappa )$ and let $\boldsymbol {F}:=(F_{n})_{n=1}^{\infty }$ be an S-Følner sequence of measurable subsets in Y. Let $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ be two sequences of positive reals. For each $n\in \Bbb N$ , we set

$$ \begin{align*} f_{n}:=1+(\unicode{x3bb}_{n}-1)\cdot 1_{F_{n}}. \end{align*} $$

Define two measures $\nu _{n}$ and $\mu _{n}$ on Y by setting: $\nu _{n}:=({a_{n}}/{\kappa (F_{n})})\,\kappa $ and $\mu _{n}\sim \nu _{n}$ with ${d\mu _{n}}/{d\nu _{n}}:=f_{n}$ . Then, $\nu _{n}(F_{n})=a_{n}$ and $\mu _{n}(F_{n})=\unicode{x3bb} _{n}a_{n}$ . We now let

$$ \begin{align*} (X,\mathfrak{B},\nu):=\bigsqcup_{n=1}^{\infty}(Y,\mathfrak{Y},\nu_{n})\quad\text{and}\quad \mu:=\bigsqcup_{n=1}^{\infty}\mu_{n}. \end{align*} $$

Denote by $T=(T_{g})_{g\in G}$ the G-action on X whose restriction to every copy of Y in X is S. Then, $\nu $ and $\mu $ are $\sigma $ -finite measures on $(X,\mathfrak {B})$ , $\nu \circ T_{g}=\nu $ and $\mu \circ T_{g}\sim \mu $ for each $g\in G$ .

Lemma 3.1. $T_{g}\in \textrm {{Aut}}_{1}(X,\mu )$ if and only if

(3.1) $$ \begin{align} \sum_{n\in\Bbb N}a_{n}|\unicode{x3bb}_{n}-1|\frac{\kappa (S_{g}F_{n}\triangle F_{n})}{\kappa(F_{n})}<\infty. \end{align} $$

Moreover, $\chi (T_{g})=0$ whenever $T_{g}\in \textrm {{Aut}}_{1}(X,\mu )$ .

Proof. We compute that

$$ \begin{align*} \int_{X}\bigg|\frac{d\mu\circ T_{g}^{-1}}{d\mu}-1\bigg|\,d\mu&= \sum_{n\in\Bbb N}\int_{Y}|f_{n}\circ S_{g}^{-1}- f_{n}|\frac {a_{n}}{\kappa(F_{n})}\,d\kappa\\ &= \sum_{n\in\Bbb N}\int_{Y}|\unicode{x3bb}_{n}-1|\cdot |1_{F_{n}}\circ S_{g}^{-1}-1_{F_{n}}|\frac{a_{n}}{\kappa(F_{n})}\,d\kappa\\ &= \sum_{n\in\Bbb N}a_{n}|\unicode{x3bb}_{n}-1|\frac{\kappa (S_{g}F_{n}\triangle F_{n})}{\kappa(F_{n})}. \end{align*} $$

Hence, (3.1) follows. In a similar way, we obtain that

$$ \begin{align*} \chi(T_{g})=\int_{X}\bigg(\frac{d\mu\circ T_{g}^{-1}}{d\mu}-1\bigg)\,d\mu= \sum_{n\in\Bbb N}\frac{a_{n}}{\nu(F_{n})}\int_{Y}(f_{n}\circ S_{g}^{-1}- f_{n})\,d\nu=0.\\[-3.8pc] \end{align*} $$

For each $\unicode{x3bb}>0$ , we let $c(\unicode{x3bb} ) :=\unicode{x3bb} ^{3}-\unicode{x3bb} +\unicode{x3bb} ^{-2}-1 = (1-\unicode{x3bb} ^{2})(1-\unicode{x3bb} ^{3})\unicode{x3bb} ^{-2}\ge 0.$

Lemma 3.2. $({d\mu }/({d\mu \circ T_{g}^{-1}}))^{2}-1\in L^{1}(X,\mu )$ if and only if

$$ \begin{align*} \sum_{n=1}^{\infty} a_{n} \frac{|1-\unicode{x3bb}_{n}^{2}|(1+\unicode{x3bb}_{n}^{3})}{\unicode{x3bb}_{n}^{2}}\frac{\kappa (S_{g}F_{n}\triangle F_{n}) }{\kappa(F_{n})} <\infty \end{align*} $$

and

$$ \begin{align*} \int_{X}\bigg(\bigg(\frac{d\mu}{d\mu\circ T_{g}^{-1}}\bigg)^{2}-1\bigg)\,d\mu=0.5\sum_{n=1}^{\infty} a_{n}c(\unicode{x3bb}_{n})\frac{\kappa (S_{g}F_{n}\triangle F_{n}) }{\kappa(F_{n})}. \end{align*} $$

Proof. We observe that

$$ \begin{align*} \int_{X}\bigg|\bigg(\frac{d\mu}{d\mu\circ T_{g}^{-1}}\bigg)^{2}-1\bigg|\,d\mu &=\sum_{n=1}^{\infty} \frac{a_{n}}{\kappa(F_{n})}\int_{Y}\bigg|\frac{f^{3}}{(f\circ T_{g}^{-1})^{2}}-f\bigg|\,d\kappa\\ &=\sum_{n=1}^{\infty} a_{n} \frac{|1-\unicode{x3bb}_{n}^{2}|(1+\unicode{x3bb}_{n}^{3})}{\unicode{x3bb}_{n}^{2}}\frac{\kappa (S_{g}F_{n}\triangle F_{n}) }{\kappa(F_{n})} \end{align*} $$

and the first claim of Lemma 3.2 follows. In a similar way,

$$ \begin{align*} \int_{X}\bigg(\bigg(\frac {d\mu}{d\mu\circ T_{g}^{-1}}\bigg)^{2}-1\bigg)\,d\mu &=\sum_{n=1}^{\infty} \frac{a_{n}}{\kappa(F_{n})}( (\unicode{x3bb}_{n}^{3}-\unicode{x3bb}_{n})\kappa(F_{n}\setminus S_{g}F_{n})\\ &\quad+(\unicode{x3bb}_{n}^{-2}-1)\kappa(S_{g} F_{n}\setminus F_{n}))\\ &=\sum_{k=1}^{\infty} a_{n}c(\unicode{x3bb}_{n})\frac{\kappa (S_{g}F_{n}\triangle F_{n}) }{2\kappa(F_{n})}, \end{align*} $$

as desired.

Let $(Y^{*},\mathfrak {Y}^{*},\mu _{n}^{*},S^{*})$ denote the Poisson suspension of the non-singular dynamical system $(Y,\mathfrak {Y},\mu _{n},S)$ , $n\in \Bbb N$ . Suppose that (3.1) holds. Then, the Poisson suspension $T^{*}=(T_{g}^{*})_{g\in G}$ of T is well defined as a non-singular G-action on a standard probability space $(X^{*},\mathfrak {B}^{*},\mu ^{*})$ . By Fact 2.18(v), we have that $(X^{*},\mu ^{*},T^{*})$ is canonically isomorphic to the direct product $\bigotimes _{n\in \Bbb N}(Y^{*},\mu _{n}^{*},S^{*})$ . Since ${d\mu _{n}}/{d\nu _{n}}-1=f_{n}-1=(\unicode{x3bb} _{n}-1)1_{F_{n}}\in L^{1}(Y,\nu _{n})$ , we obtain that $\mu _{n}^{*}\sim \nu _{n}^{*}$ . Moreover, $\nu _{n}^{*}$ is invariant under $S^{*}$ . Thus, we have shown that $(X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ is IDPFT.

Proposition 3.3. (Cf. [Da, Corollary C])

If there are $c>0$ and an S-Følner sequence $(B_{n})_{n=1}^{\infty }$ in $(Y,\mathfrak {Y},\kappa )$ such that $\kappa (B_{n})=c$ for each $n\in \Bbb N$ , then $T^{*}$ is not strongly ergodic.

Proof. Since $(X^{*},\mu ^{*},T^{*})$ is isomorphic to $\bigotimes _{n\in \Bbb N}(Y^{*},\mu _{n}^{*},S^{*})$ and $\mu _{n}^{*}\sim \nu _{n}^{*}$ for each $n\in \Bbb N$ , it follows that $(Y^{*},\nu _{1}^{*},S^{*})$ is a factor of $(X^{*},\mu ^{*},T^{*})$ . Hence, it suffices to show that $(Y^{*},\nu _{1}^{*},S^{*})$ is not strongly ergodic. We note that $\nu _{1}^{*}([B_{n}]_{0})=e^{-\nu _{1}(B_{n})}$ and

$$ \begin{align*} \nu_{1}^{*}([B_{n}]_{0}\cap S_{g}^{*}[B_{n}]_{0})=\nu_{1}^{*}([B_{n}\cup S_{g}B_{n}]_{0})=e^{-\nu_{1}(B_{n}\cup S_{g}B_{n})} \end{align*} $$

for each $n\in \Bbb N$ . Since $(B_{n})_{n=1}^{\infty }$ is S-Følner, $\sup _{g\in K}|{\nu _{1}(B_{n}\cup S_{g}B_{n})}/{\nu _{1}(B_{n})}-1|\to 0$ as $n\to \infty $ for each compact subset $K\subset G$ . Hence, $\sup _{g\in K}|\nu _{1}(B_{n}\cup S_{g}B_{n})-\nu _{1}(B_{n})|\to 0$ as $n\to \infty $ . It follows that $([B_{n}]_{0})_{n=1}^{\infty }$ is an asymptotically invariant sequence in $(Y^{*},\mathfrak {Y}^{*},\nu _{1}^{*})$ . Since $\nu _{1}(B_{n})={a_{1}c}/{\kappa (F_{1})}$ , the sequence $([B_{n}]_{0})_{n=1}^{\infty }$ is non-trivial. Hence, $T^{*}$ is not strongly ergodic.

Proposition 3.4. Let S be effective, that is, $S_{g}\ne \textrm {{Id}}$ for each $g\in G$ . If there is a measure-preserving transformation Q of $(Y,\mathfrak {Y},\kappa )$ such that:

1. (i) $QS_{g}=S_{g}Q$ for each $g\in G$ ; and

2. (ii) if $QA=A$ for a subset $A\in \mathfrak {Y}$ , then either $\kappa (A)=0$ or $\kappa (A)=+\infty $ ,

then $T^{*}$ is free.

Proof. Since $(X^{*},\mu ^{*},T^{*})=\bigotimes _{n=1}^{\infty } (Y^{*},\mu _{n}^{*},S^{*})$ and $\mu _{1}^{*}\sim \nu _{1}^{*}$ , by Fact 2.18(ii), it suffices to show that the action $S^{*}$ on $(Y^{*},\nu _{1}^{*})$ is free. Since Q preserves $\nu _{1}$ , it follows that the Poisson suspension $Q^{*}$ of Q preserves $\nu _{1}^{*}$ . We deduce from (ii) that $Q^{*}$ is ergodic. Moreover, (i) yields that $Q^{*}$ commutes with $S_{g}^{*}$ for each $g\in G$ .

Denote by $\mathcal {G}$ the space of all closed subgroups of G and endow $\mathcal {G}$ with the Fell topology [Fe]. Then, $\mathcal {G}$ is a compact metric space [Fe]. Given $\omega \in Y^{*}$ , let $G_{\omega }:=\{g\in G\mid S^{*}_{g}\omega =\omega \}$ stand for the stability group of $S^{*}$ at $\omega $ . Then, $G_{\omega }\in \mathcal {G}$ at each $\omega $ [AuMo, I, Proposition 3.7]. The mapping $ \eta :Y^{*}\ni \omega \mapsto G_{\omega }\in \mathcal {G} $ is Borel [AuMo, II, Proposition 2.3]. Since $Q^{*}$ commutes with $S^{*}$ , it is straightforward to verify that $\eta $ is invariant under $Q^{*}$ . As $Q^{*}$ is ergodic, we obtain that $\eta $ is constant. Thus, there is a subgroup $H\in \mathcal {G}$ such that $G_{\omega }=H$ at a.e. $\omega \in Y^{*}$ . Therefore, $S^{*}_{g}=\text {Id}$ for each $g\in H$ . This implies that $S_{g}=\text {Id}$ for each $g\in H$ . Since S is effective, H is trivial. Thus, we obtain that $S^{*}$ (and hence $T^{*}$ ) is free, as desired.

4 Type $II_{\infty }$ non-singular Poisson suspensions

In this section, we prove the implications (1) $\Rightarrow $ (3) of Theorems A and B for $K=II_{\infty }$ . Thus, we assume that G does not have property (T). Then, there exists a measure-preserving G-action $S=(S_{g})_{g\in G}$ on an infinite $\sigma $ -finite standard measure space $(Y,\mathfrak {Y},\kappa )$ such that

1. (α ₁) there is an S-Følner sequence $\boldsymbol {F}=(F_{n})_{n=1}^{\infty }$ such that $\kappa (F_{n})=1$ for each $n\in \Bbb N$ and

2. (α ₂) the corresponding unitary Koopman representation $U_{S}$ of G is weakly mixing

(see [Jo] or [Da, Theorem D(ii)]). We can assume, without loss of generality, that S is free. Indeed, let $L=(L_{g})_{g\in G}$ stand for the G-action on G by left translations. We endow G with the left Haar measure. Then, L is an infinite measure-preserving free G-action. The Poisson suspension $L^{*}=(L^{*}_{g})_{g\in G}$ of L is a free mixing probability preserving G-action [OrWe]. If S is not free, then we replace S with the product G-action $(S_{g}\times L^{*}_{g})_{g\in G}$ , which is free and satisfies $(\alpha _{1})$ and $(\alpha _{2})$ .

Passing to a subsequence in $\boldsymbol {F}$ , we may (and will) also assume, without loss of generality, that

1. (α ₃) $\sum _{n=1}^{\infty }\sup _{g\in K}{\kappa (S_{g}F_{n}\triangle F_{n})} <+\infty $ for each compact $K\subset G$ .

Fix two sequences $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ of positive reals such that

1. (α ₄) $a_{n}:=1$ for all $n\in \Bbb N$ and $\sum _{n=1}^{\infty }\unicode{x3bb} _{n}<+\infty $ .

We will also need one extra property: that is,

1. (α ₅) There is a $\kappa $ -preserving transformation Q that commutes with $S_{g}$ for each $g\in G$ and is such that each Q-invariant subset is either of 0-measure or of infinite measure.

Remark 4.1. If there exists an action S satisfying $(\alpha _{1})$ – $(\alpha _{4})$ , it is easy to construct a new G-action $S^{\prime }$ satisfying $(\alpha _{1})$ – $(\alpha _{5})$ . Indeed, just put $Y^{\prime }:=Y\times \Bbb Z$ , $\kappa ^{\prime }=\kappa \otimes \delta _{\Bbb Z}$ , $F_{n}^{\prime }:=F_{n}\times \{0\}$ , $S^{\prime }_{g}:=S_{g}\otimes \text {Id}$ and $Q=\text {Id}\otimes r$ , where $\delta _{\Bbb Z}$ is the counting measure on $\Bbb Z$ and r is the translation by $1$ on $\Bbb Z$ . It is straightforward to verify that $(Y^{\prime },\kappa ^{\prime }, S^{\prime }, (F_{n}^{\prime })_{n=1}^{\infty }, Q)$ satisfy $(\alpha _{1})$ – $(\alpha _{5})$ .

Denote by $(X,\mathfrak {B},\mu , T)$ the dynamical system associated with $(Y,\mathfrak {Y},\kappa ,S)$ , $\boldsymbol {F}$ , $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ via the general construction described in §3. We deduce from Lemma 3.1, $(\alpha _{3})$ and $(\alpha _{4})$ that $T_{g}\in \text {Aut}_{1}(X,\mu )$ for each $g\in G$ . Hence, the non-singular Poisson suspension $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ of $(X,\mathfrak {B},\mu , T)$ is well defined according to Fact 2.18(i).

Theorem 4.2. Let $S=(S_{g})_{g\in G}$ be as above. Then, there is a sequence $k_{n}\to \infty $ such that the general construction of §3 applied to $(Y,\mathfrak {Y},\kappa ,S)$ , $(F_{k_{n}})_{n=1}^{\infty }$ , $(a_{k_{n}})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{k_{n}})_{n=1}^{\infty }$ yields the dynamical system $(\widetilde X,\widetilde {\mathfrak {B}},\widetilde \mu , \widetilde T)$ whose Poisson suspension $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*}, \widetilde \mu ^{*}, \widetilde T^{*})$ is free, of infinite ergodic index and of Krieger type $II_{\infty }$ , non-strongly ergodic and IDPFT. Hence, $ \widetilde T^{*}$ is amenable in the Greenleaf sense. Moreover, $ \widetilde T^{*}$ is amenable if and only if G is amenable.

Before we get to the proof of this theorem, we first state an approximation lemma.

Lemma 4.3. [Da, Proposition 3.1]

Let $R=(R_{g})_{g\in G}$ be a measure-preserving G-action on an infinite $\sigma $ -finite standard measure space $(Z,\mathfrak {Z},\tau )$ . Let $\mathfrak {Z}_{0}\subset \mathfrak {Z}$ stand for the ring of subsets of finite measure. Let $Z_{1}\subset Z_{2}\subset \cdots $ be a sequence of subsets from $\mathfrak {Z}_{0}$ such that, for each $n>0$ , there is a finite partition $\mathcal {P}_{n}$ of $Z_{n}$ into subsets of equal measure satisfying the following:

• • $\bigcup _{n=1}^{\infty } Z_{n}=Z$ ; and

• • $(\mathcal {P}_{n})_{n=1}^{\infty }$ approximates $(\mathfrak {Z}_{0},\tau )$ , that is, for each $B\in \mathfrak {Z}_{0}$ and $\epsilon>0$ , there is $N>0$ such that, if $n>N$ , then there is a $\mathcal {P}_{n}$ -measurable subset $B_{n}$ with $\tau (B\triangle B_{n})<\epsilon $ .

If, for each $n>0$ and every pair of $\mathcal {P}_{n}$ -atoms A and B, there exist a finite family $g_{1},\ldots ,g_{l}\in G$ and mutually disjoint subsets ^† $A_{1},\ldots ,A_{l}$ of A such that the subsets $R_{g_{1}}A_{1},\ldots ,R_{g_{l}}A_{l}$ are mutually disjoint, $\bigsqcup _{i=1}^{l}R_{g_{i}}A_{i}\subset B$ and $\tau (\bigsqcup _{i=1}A_{i})>0.1\tau (A)$ , then R is ergodic.

Proof of Theorem 4.2

As we noted in §3, the dynamical system $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is canonically isomorphic to the infinite direct product $\bigotimes _{n\in \Bbb N}(Y^{*},\mathfrak {Y}^{*},\mu _{n}^{*},S^{*})$ . Let $B_{n}:=[F_{n}]_{0}$ and $\boldsymbol {B}:=( B_{n})_{n=1}^{\infty }$ .

Claim A. $ \sum _{n=1}^{\infty } H^{2}({\nu _{n}^{*}(B_{n})^{-1}}\nu _{n}^{*}\restriction B_{n},\mu _{n}^{*})<\infty .$

We first recall that $\nu _{n}:=({a_{n}}/{\kappa (F_{n})})\,\kappa =\kappa $ in view of $(\alpha _{1})$ and $(\alpha _{4})$ . Therefore, $\nu _{n}(F_{n})=1$ and $\nu _{n}^{*}(B_{n})=e^{-\nu _{n}(F_{n})}=e^{-1}$ . It follows now from Fact 2.18(iii) that if $\omega \in B_{n}$ then

$$ \begin{align*} \frac{d\mu_{n}^{*}}{d\nu_{n}^{*}}(\omega)=e^{-(\unicode{x3bb}_{n}-1)\nu_{n}(F_{n})}\sum_{k=0}^{\infty}\unicode{x3bb}_{n}^{k}1_{[F_{n}]_{k}}(\omega)=e^{1-\unicode{x3bb}_{n}}. \end{align*} $$

Hence,

$$ \begin{align*} H^{2}\bigg(\frac1{\nu_{n}^{*}(B_{n})}\nu_{n}^{*}\restriction B_{n},\mu_{n}^{*}\bigg)& =1-\int_{B_{n}} \frac{1}{\sqrt{\nu_{n}^{*}(B_{n})}}\sqrt{\frac{d\mu^{*}_{n}}{d\nu_{n}^{*}}} \,d\nu_{n}^{*} \\ &=1- e^{-1/2}e^{({1-\unicode{x3bb}_{n}})/2} \\ &=1-e^{-{\unicode{x3bb}_{n}}/{2}}. \end{align*} $$

Since $\sum _{n=1}^{\infty }\unicode{x3bb} _{n}<\infty $ by $(\alpha _{4})$ , Claim A follows.

Fact 2.14(i) and Claim A yield that $\mu ^{*}$ is equivalent to the restricted infinite product of $(\nu _{n}^{*})_{n=1}^{\infty }$ with respect to $\boldsymbol {B}$ . We will denote this restricted product by $(\nu ^{*})^{\boldsymbol {B}}$ . Since $\prod _{n=1}^{\infty }\nu _{n}^{*}(B_{n})=0$ , it follows that $(\nu ^{*})^{\boldsymbol {B}}(X^{*})=\infty $ . Since

$$ \begin{align*} \sum_{n=1}^{\infty}\frac{\nu_{n}^{*}(B_{n}\triangle S_{g}^{*}B_{n})}{\nu_{n}^{*}(B_{n})}= 2e\sum_{n=1}^{\infty}(\nu_{n}^{*}(B_{n})-\nu_{n}^{*}(B_{n}\cap S_{g}^{*}B_{n}))= 2\sum_{n=1}^{\infty}(1-e^{1-\nu_{n}(F_{n}\cap S_{g}F_{n})}) \end{align*} $$

and

$$ \begin{align*} \sum_{n=1}^{\infty} (1-\nu_{n}(F_{n}\cap S_{g}F_{n}))=\frac12\sum_{n=1}^{\infty} \frac{\kappa(F_{n}\triangle S_{g}F_{n})}{\kappa(F_{n})}, \end{align*} $$

it follows from this and $(\alpha _{3})$ that

(4.1) $$ \begin{align} \sup_{g\in K}\sum_{n=1}^{\infty}\frac{\nu_{n}^{*}(B_{n}\triangle S_{g}^{*}B_{n})}{\nu_{n}^{*}(B_{n})}<\infty\quad\text{ for each compact }K\subset G\text{.} \end{align} $$

Therefore, by Fact 2.11, $T^{*}$ preserves $(\nu ^{*})^{\boldsymbol {B}}$ .

Thus, $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ admits an equivalent invariant infinite $\sigma $ -finite measure $(\nu ^{*})^{\boldsymbol {B}}$ . Since S is free, S is effective. Therefore, Proposition 3.4 and $(\alpha _{5})$ yield that $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is free.

Unfortunately, we do not know whether $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is ergodic or not. However, we observe that, for each increasing sequence of positive integers $(k_{n})_{n=1}^{\infty }$ , the conditions $(\alpha _{1})$ – $(\alpha _{5})$ hold if we replace $\boldsymbol {F}$ , $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ with the subsequences $\boldsymbol {F}^{\prime }:=(F_{k_{n}})_{n=1}^{\infty }$ , $(a_{k_{n}})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{k_{n}})_{n=1}^{\infty }$ , respectively. Hence, given $(Y,\mathfrak {Y},\kappa , S)$ and these three subsequences, the general construction from §3 yields a new dynamical system which we denote by $(\widetilde X,\widetilde {\mathfrak {B}},\widetilde \mu ,\widetilde T)$ . As we have shown above, $(\alpha _{1})$ – $(\alpha _{5})$ imply that the non-singular Poisson suspension $(\widetilde X^{*},{\widetilde {\mathfrak {B}}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ of $(\widetilde X,\widetilde {\mathfrak {B}},\widetilde \mu ,\widetilde T)$ is well defined. Moreover, the dynamical system $(\widetilde X^{*},{\widetilde {\mathfrak {B}}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ is free and IDPFT. Of course, $(\widetilde X^{*},{\widetilde {\mathfrak {B}}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ is canonically isomorphic to $\bigotimes _{n\in \Bbb N}(Y^{*},\mathfrak {Y}^{*},\mu _{k_{n}}^{*},S^{*})$ . If we let $\boldsymbol B^{\prime }:=(B_{k_{n}})_{n=1}^{\infty }$ , then the measure $(\nu ^{*})^{\boldsymbol B^{\prime }}$ is infinite, $\sigma $ -finite, $\widetilde \mu ^{*}$ -equivalent and $\widetilde T^{*}$ -invariant.

Claim B. There is an increasing sequence $(k_{n})_{n=1}^{\infty }$ of positive integers such that $\bigotimes _{n\in \Bbb N}(Y^{*},\mathfrak {Y}^{*},\mu _{k_{n}}^{*},S^{*})$ is of infinite ergodic index.

To simplify the argument, we only show how to choose $(k_{n})_{n=1}^{\infty }$ so that the Cartesian square of $\bigotimes _{n\in \Bbb N}(Y^{*},\mathfrak {Y}^{*},\mu _{k_{n}}^{*},S^{*})$ is ergodic. The sequence $(k_{n})_{n=1}^{\infty }$ will be defined inductively. First, we set $k_{1}:=1$ . Suppose that we have already specified $(k_{j})_{j=1}^{n}$ . Our purpose is to define $k_{n+1}$ . For an arbitrary $j\in \Bbb N$ , let $U_{j,S^{*}}$ denote the Koopman unitary representation of G associated with the probability preserving system $(Y^{*},\nu _{j}^{*}, S^{*})$ . We deduce from $(\alpha _{2})$ and Facts 2.2 and 2.21 that the unitary representation $U_{j,S^{*}}\restriction L^{2}_{0}(Y^{*},\nu _{j}^{*})$ of G is weakly mixing. Hence, the dynamical system $(Y^{*},\nu _{j}^{*}, S^{*})$ is weakly mixing by Fact 2.4(ii). It follows that

1. (⊳) the dynamical system $\bigotimes _{l=1}^{n}(Y^{*}\times Y^{*},\nu _{k_{l}}^{*}\otimes \nu _{k_{l}}^{*}, (S^{*}_{g}\times S^{*}_{g})_{g\in G})$ is ergodic.

Let $(\mathcal {P}_{j,l})_{l=1}^{\infty }$ be a sequence of finite partitions of $Y^{*}$ into Borel subsets of equal measure $\nu _{j}^{*}$ such that

• (∘) $(\mathcal {P}_{j,l})_{l=1}^{\infty }$ approximates $(\mathfrak {Y}^{*},\nu _{j}^{*})$ .

It follows from $(\triangleright )$ that there is a finite subset $H_{n}\subset G$ such that, for every two atoms $P,Q$ of the partition $\bigotimes _{j=1}^{n}(\mathcal {P}_{k_{j},n})^{\otimes 2}$ of $(Y^{*}\times Y^{*})^{n}$ , there is a family of measured subsets $(P_{f})_{f\in H_{n}}$ of P such that:

1. (i) $P_{f}\cap P_{h}=\emptyset $ and $(S_{f}^{*})^{\times 2n}P_{f}\cap (S_{f}^{*})^{\times 2n}P_{h}=\emptyset $ for all $f,h\in H$ with $f\ne h$ ;

2. (ii) $\bigsqcup _{f\in F}(S_{f}^{*})^{\times 2n}P_{f}\subset Q$ ; and

3. (iii) $(\bigotimes _{j=1}^{n}(\nu _{k_{j}}^{*})^{\otimes 2})(\bigsqcup _{f\in H_{n}}P_{f})>0.5(\bigotimes _{j=1}^{n}(\nu _{k_{j}}^{*})^{\otimes 2})(P)$ .

Utilizing (4.1), we can select $k_{n+1}\in \Bbb N$ large so that $k_{n+1}>k_{n}$ and

(4.2) $$ \begin{align} \min_{f\in H_{n}}\prod_{j\ge k_{n+1}}\frac{\nu_{j}^{*}(S_{f}^{*}B_{j}\cap B_{j})}{\nu_{j}^{*}(B_{j})}>0.5. \end{align} $$

Repeating this process infinitely many times, we construct the entire sequence $(k_{n})_{n=1}^{\infty }$ . Since $\widetilde \mu ^{*}\sim (\nu ^{*})^{\boldsymbol {B}^{\prime }}$ , it suffices to verify that the Cartesian square of $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},(\nu ^{*})^{{\boldsymbol {B}}^{\prime }}, \widetilde T^{*})$ is ergodic. As $\widetilde T^{*}$ preserves $(\nu ^{*})^{{\boldsymbol {B}}^{\prime }}$ , we will use Lemma 4.3 for this verification. Let

(4.3) $$ \begin{align} \widetilde {\boldsymbol{B}}^{n}:=(Y^{*})^{n}\times B_{k_{n+1}}\times B_{k_{n+2}}\times\cdots\subset \widetilde X^{*}\quad\text{for each }n\in\Bbb N\text{.} \end{align} $$

For each pair of atoms $P,Q\in \bigotimes _{j=1}^{n}(\mathcal {P}_{k_{j},n})^{\otimes 2}$ and $f\in H_{n}$ , we set

$$ \begin{align*} P^{\prime} &:=P\times (B_{k_{n+1}}\times B_{k_{n+2}}\times\cdots)^{\times 2}\in (\mathcal{P}_{n}^{\prime})^{\otimes 2},\\ Q^{\prime} &:=Q\times(B_{k_{n+1}}\times B_{k_{n+2}}\times\cdots)^{\times 2}\in (\mathcal{P}_{n}^{\prime})^{\otimes 2} \quad \text{and}\\ P_{f}^{\prime} &:=P_{f}\times((B_{k_{n+1}}\cap (S^{*}_{f})^{-1}B_{k_{n+1}})\times (B_{k_{n+2}}\cap (S^{*}_{f})^{-1}B_{k_{n+2}})\times\cdots)^{\times 2}, \end{align*} $$

where $(P_{f})_{f\in H_{n}}$ are mutually disjoint subsets of P satisfying $(i)$ – $(iii)$ . Then, $\mathcal {P}_{n}^{\prime }:=(P^{\prime })_{P\in \bigotimes _{j=1}^{n}(\mathcal {P}_{k_{j},n})^{\otimes 2}}$ is a finite partition of $\widetilde {\boldsymbol {B}}^{n}\times \widetilde {\boldsymbol {B}}^{n}$ into subsets of equal measure $(\nu ^{*})^{{\boldsymbol {B}}^{\prime }}\otimes (\nu ^{*})^{{\boldsymbol {B}}^{\prime }}$ . Moreover, $(\circ )$ yields that

• (•) $(\mathcal {P}_{n}^{\prime })_{n=1}^{\infty }$ approximates $((\widetilde {\mathfrak {B}}^{*}\otimes \widetilde {\mathfrak {B}}^{*})_{0},(\nu ^{*})^{{\boldsymbol {B}}^{\prime }}\otimes (\nu ^{*})^{{\boldsymbol {B}}^{\prime }})$ ,

where $(\widetilde {\mathfrak {B}}^{*}\otimes \widetilde {\mathfrak {B}}^{*})_{0}$ denotes the ring of subsets of finite measure in $\widetilde {\mathfrak {B}}^{*}\otimes \widetilde {\mathfrak {B}}^{*}$ . We also note that $(P^{\prime }_{f})_{f\in H_{n}}$ are mutually disjoint subsets of $P^{\prime }$ and $(( \widetilde T_{f}^{*}\times \widetilde T_{f}^{*})P^{\prime }_{f})_{f\in H_{n}}$ are mutually disjoint subsets of $Q^{\prime }$ . It follows from (4.2) that, for each $f\in H_{n}$ ,

$$ \begin{align*} \prod_{j\ge n+1}\frac{\nu_{k_{j}}^{*}(S_{f}^{*}B_{k_{j}}\cap B_{k_{j}})}{\nu_{k_{j}}^{*}(B_{k_{j}})}>0.5. \end{align*} $$

Using this inequality and $(iii)$ , we obtain that

$$ \begin{align*} ((\nu^{*})^{{\boldsymbol{B}}^{\prime}}\otimes (\nu^{*})^{{\boldsymbol{B}}^{\prime}})\bigg(\bigsqcup_{f\in H_{n}}P^{\prime}_{f}\bigg)&> \frac{1/2(\bigotimes_{j=1}^{n}(\nu_{k_{j}}^{*})^{\otimes 2})(P)}{(\prod_{j=1}^{n}\nu^{*}_{k_{j}}(B_{k_{j}}))^{2}}\cdot \frac14 \\ &=\frac 18 ((\nu^{*})^{{\boldsymbol{B}}^{\prime}}\otimes (\nu^{*})^{{\boldsymbol{B}}^{\prime}})(P^{\prime}) \end{align*} $$

for every pair of $\mathcal {P}_{n}^{\prime }$ -atoms $P^{\prime },Q^{\prime }$ . Therefore, it follows from Lemma 4.3 that the G-action $(\widetilde T_{g}^{*}\times \widetilde T_{g}^{*})_{g\in G}$ on $(\widetilde {X}^{*}\otimes \widetilde {X}^{*},(\nu ^{*})^{{\boldsymbol {B}}^{\prime }}\otimes (\nu ^{*})^{{\boldsymbol {B}}^{\prime }})$ is ergodic. Thus, Claim B is proved.

It follows that $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ is of type $II_{\infty }$ . We deduce from Proposition 3.3 and $(\alpha _{1})$ that $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ is not strongly ergodic. Since $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ is IDPFT, Proposition 2.25 yields that $T^{*}$ is amenable in the Greenleaf sense. Moreover, $\widetilde T^{*}$ is amenable if and only if G is amenable.

Thus, Theorem 4.2 is proved completely.

Theorem 4.4. If G is Haagerup, then we can choose the system $(\widetilde X,\widetilde {\mathfrak {B}},\widetilde \mu , \widetilde T)$ in the statement of Theorem 4.2 in such a way that the Poisson suspension $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*}, \widetilde \mu ^{*}, \widetilde T^{*})$ is of 0-type (in addition to the other properties of $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*}, \widetilde \mu ^{*}, \widetilde T^{*})$ listed in Theorem 4.2).

Proof. In view of [DeJoZu] (see also [Da]), there is an infinite $\sigma $ -finite measure-preserving system $(Y,\mathfrak {Y},\kappa , S)$ of 0-type satisfying $(\alpha _{1})$ – $(\alpha _{5})$ . Then, we repeat the proof of Theorem 4.2 verbatim to obtain a non-strongly ergodic free IDPFT system $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ of infinite ergodic index and of type $II_{\infty }$ . The $\widetilde \mu ^{*}$ -equivalent infinite $\sigma $ -finite $\widetilde T^{*}$ -invariant measure is $(\nu ^{*})^{{\boldsymbol {B}}^{\prime }}$ . Since the dynamical system $(Y,\nu _{n},S)$ is of 0-type, the Poisson suspension $(Y^{*},\nu _{n}^{*},S^{*})$ is mixing in view of Fact 2.21 for each $n\in \Bbb N$ . Then, by Corollary 2.13, $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},(\nu ^{*})^{{\boldsymbol {B}}^{\prime }},\widetilde T^{*})$ is of 0-type. Hence, $(\widetilde X^{*},\widetilde {\mathfrak {B}}^{*},\widetilde \mu ^{*},\widetilde T^{*})$ is of 0-type too.

5 Type $III_{0}$ ergodic Poisson suspensions

Our purpose in this section is to prove the implications (1) $\Rightarrow $ (3) from Theorems A and B for $K=III_{0}$ . Thus, we assume that G does not have property (T). As in §4, we fix a measure-preserving free G-action $S=(S_{g})_{g\in G}$ on an infinite $\sigma $ -finite standard measure space $(Y,\mathfrak {Y},\kappa )$ such that:

1. (α ₁) there is an S-Følner sequence $\boldsymbol {F}=(F_{n})_{n=1}^{\infty }$ such that $\kappa (F_{n})=1$ for each $n\in \Bbb N$ ;

2. (α ₂) the corresponding unitary Koopman representation $U_{S}$ of G is weakly mixing;

3. (α ₃) $\sum _{n=1}^{\infty }\sup _{g\in K}{\kappa (S_{g}F_{n}\triangle F_{n})}<+\infty $ for each compact $K\subset G$ ; and

4. (α ₄) there is a $\kappa $ -preserving transformation Q that commutes with $S_{g}$ for each $g\in G$ and such that each Q-invariant subset is either of 0-measure or of infinite measure.

We deduce from $(\alpha _{2})$ and Fact 2.2 that there is a dispersed sequence $(g_{n})_{n=1}^{\infty }$ in G such that

1. (α ₅) S is of $0$ -type along the subset $H:=\{g_{n}g_{m}^{-1}\mid n>m, n,m\in \Bbb N\}\subset G$ .

Utilizing $(\alpha _{3})$ and passing to a suitable subsequence in $\boldsymbol {F}$ , which we denote by the same symbol $\boldsymbol {F}$ , we may (and will) assume, without loss of generality, that

(5.1) $$ \begin{align} \sum_{n=1}^{\infty}\max_{1\le l\le n}{\kappa(S_{g_{l}}F_{n}\triangle F_{n})} \le 2. \end{align} $$

Let $(l_{n})_{n=1}^{\infty }$ be a sequence of positive integers such that $l_{1}|l_{2}$ , $l_{2}|l_{3},\ldots $ , $\lim _{n\to \infty } l_{n}=+\infty $ and $\sum _{n=1}^{\infty }(1/{n4^{l_{n}}})=+\infty $ . We now set

1. (α ₆ ) $\unicode{x3bb} _{n}:=2^{l_{n}}$ and $a_{n}:=c(\unicode{x3bb} _{n})^{-1}n^{-1}$ for each $n>0$ .

Denote by $(X,\mathfrak {B},\mu , T)$ the dynamical system associated with $(Y,\mathfrak {Y},\kappa ,S)$ , $\boldsymbol {F}$ , $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ via the general construction from §3. Since $a_{n}\sim \unicode{x3bb} _{n}^{-3}n^{-1}$ as $n\to \infty $ , it follows from $(\alpha _{3})$ and $(\alpha _{6})$ that, for each $g\in G$ ,

(5.2) $$ \begin{align} &\sum_{n\in\Bbb N}a_{n}|\unicode{x3bb}_{n}-1|\frac{\kappa (S_{g}F_{n}\triangle F_{n})}{\kappa(F_{n})}<\infty\quad\text{and} \end{align} $$

(5.3) $$ \begin{align} &\sum_{n=1}^{\infty} a_{n}\frac{ |1-\unicode{x3bb}_{n}^{2}|(1+\unicode{x3bb}_{n}^{3})}{\unicode{x3bb}_{n}^{2}} \frac{\kappa(S_{g}F_{n}\triangle F_{n})}{\kappa(F_{n})}<\infty. \end{align} $$

Hence, (5.2) and Lemma 3.1 yield that $T_{g}\in \text {Aut}_{1}(X,\mu )$ and $\chi (T_{g})=0$ for each $g\in G$ . Therefore, as was explained in §3, the non-singular Poisson suspension $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ of $(X,\mathfrak {B},\mu , T)$ is well defined.

Theorem 5.1. The dynamical system $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is weakly mixing non-strongly ergodic IDPFT of Krieger type $III_{0}$ . Hence, $T^{*}$ is amenable in the Greenleaf sense. Moreover, $T^{*}$ is amenable if and only if G is amenable.

Proof. It was shown in §3 that $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is IDPFT. Hence, $T^{*}$ is amenable in the Greenleaf sense by Proposition 2.25. Proposition 3.4 and $(\alpha _{4})$ imply that $T^{*}$ is free. Proposition 3.3 and $(\alpha _{1})$ imply that $T^{*}$ is not strongly ergodic.

We now prove that $T^{*}$ is weakly mixing. We note that (5.3), Lemma 3.2 and $(\alpha _{6})$ yield that

$$ \begin{gather*} \bigg(\frac {d\mu}{d\mu\circ T_{g}^{-1}}\bigg)^{2}-1\in L^{1}(X,\mu)\quad\text{and}\\ \int_{X}\bigg(\bigg(\frac{d\mu}{d\mu\circ T_{g}^{-1}}\bigg)^{2}-1\bigg)\,d\mu=\sum_{n=1}^{\infty} a_{n}c(\unicode{x3bb}_{n})\frac{\kappa (S_{g}F_{n}\triangle F_{n}) }{2\kappa(F_{n})}=\sum_{n=1}^{\infty}\frac1{2n}{\kappa (S_{g}F_{n}\triangle F_{n}) } \end{gather*} $$

for every $g\in G$ . Hence, for each $k\in \Bbb N$ ,

$$ \begin{align*} \int_{X}\bigg(\bigg(\frac{d\mu}{d\mu\circ T_{g_{k}}^{-1}}\bigg)^{2}-1\bigg)\,d\mu \le\frac 12\sum_{n=1}^{k}\frac 1n+ \sum_{n=k+1}^{\infty}{\kappa ( S_{g_{k}} F_{n}\triangle F_{n} )}\le \frac 12\log k+3 \end{align*} $$

in view of (5.1). This implies that

$$ \begin{align*} \sum_{k=1}^{\infty} \frac 1{k^{2}}\,e^{\int_{X}(({d\mu}/({d\mu\circ T_{g_{k}}^{-1}}))^{2}-1)\,d\mu}<+\infty. \end{align*} $$

Therefore, by Proposition 2.20,

$$ \begin{align*} \sum_{k=1}^{\infty}\frac{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}{d\mu^{*}}(\omega)=+\infty\quad\text{at a.e. }\omega\text{.} \end{align*} $$

On the other hand, in view of Fact 2.21, $(Y^{*},\mathfrak {Y}^{*},\nu _{k}^{*}, S^{*})$ is mixing along H for each $k\in \Bbb N$ because $(\alpha _{5})$ holds. (We refer to §3 for the definitions of $(\nu _{k})_{k=1}^{\infty }$ and $(\mu _{k})_{k=1}^{\infty }$ .) Hence, by Proposition 2.16, $T^{*}$ is weakly mixing.

Claim A. The dynamical system $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is of type $III$ .

Suppose, by contraposition, that $T^{*}$ is of type $II$ . Let $\eta $ stand for a $T^{*}$ -invariant $\sigma $ -finite measure equivalent to $\mu ^{*}$ . Fix $n\in \Bbb N$ . Since $\bigotimes _{k=1}^{n}\mu _{k}^{*}\sim \bigotimes _{k=1}^{n}\nu _{k}^{*}$ , it follows that the projection of $\eta $ to $(Y^{*})^{n}$ along the mapping

$$ \begin{align*} \pi_{n}:X^{*}\ni(x^{*}_{k})_{k=1}^{\infty}\mapsto (x^{*}_{k})_{k=1}^{n}\in( X^{*})^{n} \end{align*} $$

has the same collection of 0-measure subsets as the measure $\bigotimes _{k=1}^{n}\nu _{k}^{*}$ . Hence, we can disintegrate $\eta $ over $\bigotimes _{k=1}^{n}\nu _{k}^{*}$ along $\pi _{n}$ . Thus, there is measurable field $(Y^{*})^{n}\ni y\mapsto \eta _{y}$ of $\sigma $ -finite measures on $(Y^{*})^{\Bbb N}$ such that

(5.4) $$ \begin{align} \eta=\int_{(Y^{*})^{n}}\delta_{y}\otimes\eta_{y}\,d\bigg(\bigotimes_{k=1}^{n}\nu_{k}^{*}\bigg)(y). \end{align} $$

Since Q preserves $\nu _{k}$ , we obtain that the Poisson suspension $Q^{*}$ of Q preserves $\nu _{k}^{*}$ . It follows from $(\alpha _{4})$ that $Q^{*}$ is ergodic with respect to $\nu _{k}^{*}$ for each $k\in \Bbb N$ . Moreover, $Q^{*}$ commutes with $S^{*}$ . Let

$$ \begin{align*} \Gamma_{n}:=\{(Q^{*})^{l_{1}}\otimes\cdots\otimes (Q^{*})^{l_{n}}\mid l_{1},\ldots,l_{n}\in\Bbb Z\}. \end{align*} $$

Then, $\Gamma _{n}$ is a countable group (isomorphic to $\Bbb Z^{n}$ ) of measure-preserving transformations of the product space $((Y^{*})^{n},(\mathfrak {Y}^{*})^{\otimes n},\bigotimes _{k=1}^{n}\nu _{k}^{*})$ . Of course, $\Gamma _{n}\otimes 1$ is a group of $\mu ^{*}$ -non-singular transformations commuting with $T^{*}$ . Since $\eta \sim \mu ^{*}$ , $\Gamma _{n}\otimes 1$ is also $\eta $ -non-singular. Since $T^{*}$ is ergodic and $\eta $ -preserving, it follows that $\eta \circ \gamma =a_{\gamma }\eta $ for some $a_{\gamma }>0$ , for each transformation $\gamma \in \Gamma _{n}\otimes 1$ . Since $\gamma $ preserves a finite measure $(\bigotimes _{k=1}^{n}\nu _{k}^{*})\otimes \bigotimes _{k>n}\mu _{n}^{*}$ which is equivalent to $\eta $ , we obtain that $\gamma $ is conservative with respect to $\eta $ . Hence, $a_{\gamma }=1$ , that is, $\gamma $ preserves $\eta $ for each $\gamma \in \Gamma _{n}\otimes 1$ . We deduce from this fact and (5.4) that the mapping

$$ \begin{align*} \Psi:(Y^{*})^{n}\ni y\mapsto \eta_{y} \end{align*} $$

is invariant under $\Gamma _{n}$ . Since $\Gamma _{n}$ is ergodic with respect to $\bigotimes _{k=1}^{n}\nu _{k}^{*}$ , it follows that $\Psi $ is constant almost everywhere, that is, there exists a $\sigma $ -finite measure $\eta ^{n}$ on $({({Y}^{*})}^{\Bbb N},(\mathfrak {Y}^{*})^{\otimes \Bbb N}),$ such that $\eta _{y}=\eta ^{n}$ at a.e. $y\in (Y^{*})^{n}$ . We thus obtain that $\eta =(\bigotimes _{k=1}^{n}\nu _{k}^{*})\otimes \eta ^{n}$ . Since n is arbitrary, $\eta $ is an Moore–Hill product of $(\nu _{k}^{*})_{k=1}^{\infty }$ (see Definition 2.10(ii)). Hence, by Fact 2.14(ii), $\eta $ is proportional to the restricted product of $(\nu ^{*}_{k})_{k=1}^{\infty }$ with respect to some sequence $\boldsymbol {C}=(C_{n})_{n=1}^{\infty }$ of subsets $C_{n}\subset Y^{*}$ . In particular,

(5.5) $$ \begin{align} \eta(C_{1}\times C_{2}\times\cdots)>0. \end{align} $$

If $\prod _{n=1}^{\infty }\nu _{n}^{*}(C_{n})>0$ , then $\eta \sim \bigotimes _{n=1}^{\infty }\nu _{n}^{*}$ and hence $\mu ^{*}\sim \nu ^{*}$ . The latter happens if and only if $ \sqrt {{d\mu }/{d\nu }}-1\in L^{2}(X,\nu ) $ by Fact 2.18(ii). However,

$$ \begin{align*} \int_{X}\bigg(\sqrt{\frac{d\mu}{d\nu}}-1\bigg)^{2}\,d\nu=\sum_{n=1}^{\infty}(\sqrt{\unicode{x3bb}_{n}}-1)^{2}a_{n}\asymp \sum_{n=1}^{\infty}\frac1{n\unicode{x3bb}_{n}^{2}}=\sum_{n=1}^{\infty}\frac 1{n4^{l_{n}}}=+\infty \end{align*} $$

by choice of $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ , which is a contradiction. Hence, $\prod _{n=1}^{\infty }\nu _{n}^{*}(C_{n})=0$ or, equivalently, $\sum _{n=1}^{\infty }\nu _{n}^{*}(Y_{n}^{*}\setminus C_{n})=+\infty $ . Since $\eta \sim \mu ^{*}$ , we deduce from (5.5) that

$$ \begin{align*} 0<\mu^{*}(C_{1}\times C_{2}\times\cdots)=\prod_{n=1}^{\infty}\mu_{n}^{*}(C_{n}) \end{align*} $$

or, equivalently, $\sum _{n=1}^{\infty }\mu _{n}^{*}(Y_{n}^{*}\setminus C_{n})<+\infty $ . On the other hand, it follows from Fact 2.18(iii) that

$$ \begin{align*} \frac{d\nu_{n}^{*}}{d\mu_{n}^{*}}(\omega)=e^{-((1/{\unicode{x3bb}_{n}})-1)\mu_{n}(F_{n})}\unicode{x3bb}_{n}^{-\omega(F_{n})} \end{align*} $$

at $\mu _{n}^{*}$ -a.e. $\omega \in Y_{n}^{*}$ . Since $\unicode{x3bb} _{n}\ge 1$ and $\mu _{n}(F_{n})=a_{n}\unicode{x3bb} _{n}={\unicode{x3bb} _{n}}/{nc(\unicode{x3bb} _{n})}\le 1$ , we obtain that ${d\nu _{n}^{*}}/{d\mu _{n}^{*}}(\omega )\le e$ for a.e. $\omega \in X_{n}^{*}$ . Hence,

$$ \begin{align*} +\infty=\sum_{n=1}^{\infty}\nu_{n}^{*}(Y_{n}^{*}\setminus C_{n})\le e^{}\sum_{n=1}^{\infty}\mu_{n}^{*}(Y_{n}^{*}\setminus C_{n})<+\infty, \end{align*} $$

which is a contradiction. Thus, Claim A is proved.

Claim B. $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is of type $III_{0}$ .

Fix $n\in \Bbb N$ . Let $\mu ^{*,n}$ denote the infinite product measure $(\bigotimes _{k=1}^{n}\nu _{n}^{*})\otimes \bigotimes _{k>n}\mu _{k}^{*}$ . Of course, $\mu ^{*,n}\sim \mu ^{*}$ . Hence, $r(T^{*},\mu ^{*})=r(T^{*},\mu ^{*,n})$ . We recall that

$$ \begin{align*} f_{n}:=\frac{d\mu_{n}}{d\nu_{n}}=1+(\unicode{x3bb}_{n}-1)1_{F_{n}}. \end{align*} $$

From Fact 2.18(iv), we deduce that

$$ \begin{align*} \frac{d\mu_{k}^{*}\circ S_{g}^{*}}{d\mu_{k}^{*}}(\omega)=\prod_{\{y\in Y\mid \omega(\{y\})=1\}}\frac{d\mu_{k}\circ S_{g}}{d\mu_{k}}(y)= \prod_{\{y\in Y\mid \omega(\{y\})=1\}}\frac{f_{k}(S_{g}y)}{f_{k}(y)} \in\{\unicode{x3bb}_{k}^{m}\mid m\in\Bbb Z\} \end{align*} $$

for each $g\in G$ at $\mu _{k}^{*}$ -a.e. $\omega \in Y^{*}$ . We deduce from this, $(\alpha _{6})$ and the fact that $(X^{*},\mathfrak {B}^{*},\mu ^{*,n}, T^{*})$ is IDPFT that

$$ \begin{align*} \frac{d\mu^{*,n}\circ T_{g}^{*}}{d\mu^{*,n}}(\omega)=\prod_{k>n}^{\infty}\frac{d\mu_{k}^{*}\circ S_{g}^{*}}{d\mu_{k}^{*}}(\omega_{k})\in\{2^{kl_{n+1}}\mid k\in\Bbb Z\} \end{align*} $$

for each $g\in G$ at a.e. $\omega \in X^{*}$ . Hence, $r(T^{*},\mu ^{*,n})\subset \{2^{kl_{n+1}}\mid k\in \Bbb Z\}$ . This yields that $r(T^{*},\mu ^{*})\subset \bigcap _{n=1}^{\infty }\{2^{kl_{n+1}}\mid k\in \Bbb Z\}=\{1\}$ . Since $T^{*}$ is not of type $II$ , it follows that $T^{*}$ is of type $III_{0}$ . Thus, Claim B is proved.

Remark 5.2. By changing the parameters $(a_{n})_{n=1}^{\infty }$ and $(l_{n})_{n=1}^{\infty }$ in $(\alpha _{6})$ in an appropriate way we can obtain $T^{*}$ with infinite ergodic index (which is stronger than the weak mixing). For that, we choose $(l_{n})_{n=1}^{\infty }$ so that $\sum _{n=1}^{\infty }({1}/{n4^{l_{n}}\log (n+1)})=+\infty $ . Then, we let $\unicode{x3bb} _{n}:=2^{l_{n}}$ and $a_{n}:=c(\unicode{x3bb} _{n})^{-1} 1/{n\log {(n+1)}}$ for each $n\in \Bbb N$ . Let $(X,\mathfrak {B},\mu ,T)$ be the dynamical system associated with these new parameters via the general construction from §3. Then, the Poisson suspension $(X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ possesses all the properties listed in the statement of Theorem 5.1 and, in addition, $T^{*}$ is of infinite ergodic index. Indeed, fix $l>1$ . By Fact 2.18(v), the l-th Cartesian power of $(X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ is the Poisson suspension of the disjoint union $\bigsqcup _{j=1}^{l}(X,\mathfrak {B},\mu , T)$ of l copies of $(X,\mathfrak {B},\mu , T)$ . It is straightforward to verify that $\bigsqcup _{j=1}^{l}(X,\mathfrak {B},\mu , T)$ is the dynamical system associated with the disjoint union $\bigsqcup _{j=1}^{l}(Y,\mathfrak {Y},\kappa , S)$ of l copies of $(Y,\mathfrak {Y},\kappa , S)$ and the sequences $(\bigsqcup _{j=1}^{l} F_{n})_{n=1}^{\infty }$ , $(la_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ via the general construction of §3. Then, a slight modification of the proof of Theorem 5.1 yields that $(X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})^{\times l}$ is weakly mixing. We leave the details to the reader. Thus, $(X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ is of infinite ergodic index.

Theorem 5.3. If G is Haagerup, then there is a dynamical system $(X,\mathfrak {B},\mu , T)$ so that the Poisson suspension $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ is of 0-type (in addition to the other properties of $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ listed in Theorem 5.1).

Proof. Since G is Haagerup, there is a measure-preserving G-action $S=(S_{g})_{g\in G}$ on an infinite $\sigma $ -finite standard measure space $(Y,\mathfrak {Y},\kappa )$ such that $(\alpha _{1})$ – $(\alpha _{6})$ hold and, moreover, S is of 0-type (see [DeJoZu] or [Da]). Then, following the proof of Theorem 5.1 verbatim, we construct a dynamical system $( X,\mathfrak {B},\mu , T)$ whose Poisson suspension $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ possesses all the properties listed in the statement of Theorem 5.1. It remains only to prove that $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ is of 0-type. By [DaKoRo1, Proposition 6.9], $T^{*}$ is of 0-type if and only if $\int _{X}(\sqrt {({d\mu \circ T_{g}})/{d\mu }}-1)^{2}\,d\mu \to 0$ as $g\to \infty $ . Since

$$ \begin{align*} \int_{X}\bigg(\sqrt{\frac{d\mu\circ T_{g}}{d\mu}}-1\bigg)^{2}\,d\mu\le \int_{X}\bigg|{\frac{d\mu\circ T_{g}}{d\mu}}-1\bigg|\,d\mu \end{align*} $$

and, as was computed in the proof of Lemma 3.1,

$$ \begin{align*} \int_{X}\bigg|{\frac{d\mu\circ T_{g}}{d\mu}}-1\bigg|\,d\mu=\sum_{n=1}^{\infty} a_{n}|\unicode{x3bb}_{n}-1|\kappa(S_{g}F_{n}\triangle F_{n}), \end{align*} $$

it suffices to show that $\sum _{n=1}^{\infty } a_{n}|\unicode{x3bb} _{n}-1|(1-\kappa (S_{g}F_{n}\cap F_{n}))\to 0$ as $g\to \infty $ . We recall that $a_{n}\unicode{x3bb} _{n}\sim 1/{n\unicode{x3bb} _{n}^{2}}=1/{n4^{l_{n}}}$ and $\sum _{n=1}^{\infty }(1/{n4^{l_{n}}})<\infty $ . Hence, given $\epsilon>0$ , there is $N>0$ such that $ \sum _{n=N+1}^{\infty } a_{n}|\unicode{x3bb} _{n}-1|(1-\kappa (S_{g}F_{n}\cap F_{n}))<\epsilon. $ On the other hand, as S is of 0-type, $\kappa (S_{g}F_{n}\cap F_{n})\to \kappa (F_{n})^{2}=1$ for each $n=1,\ldots , N$ as $g\to \infty $ . Hence $\lim _{g\to \infty }\sum _{n=1}^{\infty } a_{n}|\unicode{x3bb} _{n}-1|(1-\kappa (S_{g}F_{n}\cap F_{n}))=0$ , as desired.

6 Type $III_{\unicode{x3bb} }$ ergodic Poisson suspensions for $\unicode{x3bb} \in (0,1)$

Fix $\unicode{x3bb} \in (0,1)$ . In this section we prove the implications (1) $\Rightarrow $ (3) of Theorems A and B for $K=III_{\unicode{x3bb} }$ . Thus, we assume that G does not have property (T).

Let $(Y,\mathfrak {Y},\kappa , S)$ , $\boldsymbol {F}=(F_{n})_{n=1}^{\infty }$ , $(g_{k})_{k=1}^{\infty }$ and let H denote the same objects as in §5. Thus, $(\alpha _{1})$ – $(\alpha _{5})$ and (5.1) from §5 hold. We now define $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ in the following way: that is,

1. $(\alpha _{6})^{\prime }$ $\unicode{x3bb} _{2n-1}:=\unicode{x3bb} _{2n}^{-1}:=\unicode{x3bb} $ and $a_{2n-1}:=\unicode{x3bb} ^{-1}a_{2n}:= 1/{(n\log (n+1))}$ for each $n\in \Bbb N$ .

Denote by $(X,\mathfrak {B},\mu , T)$ the dynamical system associated with $(Y,\mathfrak {Y},\kappa ,S)$ , $\boldsymbol {F}$ , $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ via the general construction from §3. It follows from $(\alpha _{3})$ and $(\alpha _{6})^{\prime }$ that, for each $g\in G$ ,

(6.1) $$ \begin{align} \sum_{n\in\Bbb N}a_{n}|\unicode{x3bb}_{n}-1|\frac{\kappa (S_{g}F_{n}\triangle F_{n})}{\kappa(F_{n})}<\infty\quad\text{and}\end{align} $$

(6.2) $$ \begin{align} \sum_{n=1}^{\infty} a_{n}\frac{ |1-\unicode{x3bb}_{n}^{2}|(1+\unicode{x3bb}_{n}^{3})}{\unicode{x3bb}_{n}^{2}} \frac{\kappa(S_{g}F_{n}\triangle F_{n})}{\kappa(F_{n})}<\infty. \end{align} $$

Hence, (6.1) and Lemma 3.1 yield that $T_{g}\in \text {Aut}_{1}(X,\mu )$ and $\chi (T_{g})=0$ for each $g~\in ~G$ . Therefore, the non-singular Poisson suspension $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is well defined by Fact 2.18(i).

Theorem 6.1. The dynamical system $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is weakly mixing non-strongly ergodic IDPFT of Krieger type $III_{\unicode{x3bb} }$ . Hence, $T^{*}$ is amenable in the Greenleaf sense. Moreover, $T^{*}$ is amenable if and only if G is amenable.

Proof. Following the proof of Theorem 5.1, we obtain that $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is IDPFT and hence amenable in the Greenleaf sense. Moreover, we deduce from $(\alpha _{1})$ , $(\alpha _{4})$ and Propositions 3.3 and 3.4 that $T^{*}$ is free and not strongly ergodic. It follows from (6.2) and Lemma 3.2 that $({d\mu }/({d\mu \circ T_{g}^{-1}}))^{2}-1\in L^{1}(X,\mu )$ for each $g\in G$ . Since $c(\unicode{x3bb} ^{-1})={c(\unicode{x3bb} )}/\unicode{x3bb} $ , we deduce from $(\alpha _{6})^{\prime }$ that

$$ \begin{align*} c(\unicode{x3bb}_{2n})a_{2n}=c(\unicode{x3bb}_{2n-1})a_{2n-1}=\frac{c(\unicode{x3bb})}{n\log(n+1)}\quad\text{for every }n\in\Bbb N. \end{align*} $$

Therefore, for each $k\in \Bbb N$ , in view of (5.1),

$$ \begin{align*} \int_{X}\bigg(\bigg(\frac{d\mu}{d\mu\circ T_{g_{k}}^{-1}}\bigg)^{2}-1\bigg)\,d\mu&=\sum_{n=1}^{\infty} a_{n}c(\unicode{x3bb}_{n})\frac{\kappa (S_{g}F_{n}\triangle F_{n}) }{2\kappa(F_{n})}\\ &\le \frac{1}2 \sum_{n=1}^{k}a_{n}c(\unicode{x3bb}_{n})+ \frac{c(\unicode{x3bb})}2 \sum_{n=k+1}^{\infty}{\kappa (S_{g_{n}}F_{n}\triangle F_{n}) }\\ &\le \frac14 \sum_{n=1}^{k} \frac{c(\unicode{x3bb})}{n\log(1+n/2)} +c(\unicode{x3bb})\\ &\le c(\unicode{x3bb})\log\log k +c(\unicode{x3bb}). \end{align*} $$

This implies that

$$ \begin{align*} \sum_{k=1}^{\infty} \frac 1{(k\log k)^{1.5}}\,e^{\int_{X}(({d\mu}/({d\mu\circ T_{g_{k}}^{-1}}))^{2}-1)\,d\mu}<+\infty. \end{align*} $$

By Proposition 2.20 (see (2.4)),

$$ \begin{align*} \sum_{k=1}^{\infty}\bigg(\frac{d\mu^{*}\circ (T_{g_{k}}^{*})^{-1}}{d\mu^{*}}(\omega)\bigg)^{4/3}=+\infty\quad\text{at a.e. }\omega\text{.} \end{align*} $$

On the other hand, $(Y^{*},\nu _{n}^{*}, S^{*})$ is mixing along H because $(Y,\nu _{n},S)$ is of 0-type along H for each $n\in \Bbb N$ , in view of Fact 2.21. Therefore, $(X^{*},\mu ^{*},T^{*})$ is weakly mixing in view of Proposition 2.16. (Of course, $ \sum _{k=1}^{\infty }(({d\mu ^{*}\circ (T_{g_{k}}^{*})^{-1}})/{d\mu ^{*}})(\omega )=+\infty $ at a.e. $\omega $ .) Moreover, the associated flow of $(X^{*},\mu ^{*},T^{*})$ coincides with the associated flow of $(X^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ by Proposition 2.17.

Claim A. The dynamical system $(X^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ is of type $III_{\unicode{x3bb} }$ .

We first show that $\unicode{x3bb} ^{-1}$ is an essential value of the Radon–Nikodym cocycle of $(X^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ . Since $\mu _{2k+1}(F_{2k+1})=\unicode{x3bb} _{2k+1}a_{2k+1}$ , it follows from $(\alpha _{6})^{\prime }$ that, for each $n>0$ ,

$$ \begin{align*} \sum_{k=n}^{+\infty}\mu_{2k+1}(F_{2k+1})=+\infty\quad\text{and}\quad\lim_{k\to\infty}\mu_{2k+1}(F_{2k+1})= 0. \end{align*} $$

Therefore, there is $m_{n}>n$ such that $ \alpha _{n}:=\sum _{k=n}^{m_{n}-1}\mu _{2k+1}(F_{2k+1})\in (0.5,1). $ Denote the probability space $((Y^{*})^{2(m_{n}-n)}, \mu ^{*}_{2n+1}\otimes \cdots \otimes \mu ^{*}_{2m_{n}})$ by $(X^{*}_{2n+1,2m_{n}},\mu ^{*}_{2n+1,2m_{n}})$ . We also let $\nu ^{*}_{2n+1,2m_{n}}:=\nu ^{*}_{2n+1}\otimes \cdots \otimes \nu ^{*}_{2m_{n}}$ . Since

$$ \begin{align*} \mu_{2n-1}(F_{2n-1})-\nu_{2n-1}(F_{2n-1})+\mu_{2n}(F_{2n})-\nu_{2n}(F_{2n})=0, \end{align*} $$

it follows from Fact 2.18(iii) that, for a.e. $(\omega ,\eta )\in Y^{*}\times Y^{*}$ ,

(6.3) $$ \begin{align} \frac{d(\mu_{2n-1}^{*}\otimes \mu_{2n}^{*})}{d(\nu_{2n-1}^{*}\otimes\nu_{2n}^{*})}(\omega,\eta) &=\sum_{k=0}^{\infty}\unicode{x3bb}^{k}1_{[F_{2n-1}]_{k}}(\omega) \sum_{k=0}^{\infty}\unicode{x3bb}^{-k}1_{[F_{2n}]_{k}}(\eta)\nonumber\\ &=\sum_{k=-\infty}^{\infty}\unicode{x3bb}^{k}1_{B_{n,k}}(\omega,\eta), \end{align} $$

where $1_{B_{n,k}}=\sum _{k=j-r} 1_{[F_{2n-1}]_{j}}1_{[F_{2n}]_{r}}$ . Hence, the mapping

$$ \begin{align*} \vartheta_{n}:X^{*}_{2n+1,2m_{n}}\ni \omega\mapsto\log_{\unicode{x3bb}}\frac{d\mu^{*}_{2n+1,2m_{n}}}{d\nu^{*}_{2n+1,2m_{n}}}(\omega)\in\Bbb Z \end{align*} $$

is well defined. Applying (6.3), we obtain that

$$ \begin{align*} \vartheta_{n}(\omega)=\sum_{k=n+1}^{m_{n}}(\omega(F_{2k-1})-\omega(F_{2k})),\quad\omega\in X^{*}_{2n+1,2m_{n}}. \end{align*} $$

Thus, $\mu ^{*}_{2n+1,2m_{n}}\circ \vartheta _{n}^{-1}$ is the distribution of the difference of two independent Poisson random variables $\omega \mapsto \sum _{k=n+1}^{m_{n}}\omega (F_{2k-1})$ and $\omega \mapsto \sum _{k=n+1}^{m_{n}}\omega (F_{2k})$ , one with parameter $\alpha _{n}$ and the other with parameter $\unicode{x3bb} \alpha _{n}$ . In other words, $\mu ^{*}_{2n+1,2m_{n}}\circ \vartheta _{n}^{-1}$ is the Skellam distribution with parameters $\alpha _{n},\unicode{x3bb} \alpha _{n}$ (see [AbSt]). For $i=0,1$ , let $\Delta _{i}:=\vartheta _{n}^{-1}(\{i\})\subset X^{*}_{2n+1,2m_{n}}$ . Then, ^†

$$ \begin{align*} \mu^{*}_{2n+1,2m_{n}}(\Delta_{1})= e^{-\alpha_{n}(1+\unicode{x3bb})}\sum_{k=0}^{\infty}\frac{\alpha_{n}^{k+1}(\unicode{x3bb}\alpha_{n})^{k}}{(k+1)!k!}>\frac{\alpha_{n}}{e^{\alpha_{n}(1+\unicode{x3bb})}}>\frac1{16} \end{align*} $$

and $\mu ^{*}_{2n+1,2m_{n}}(\Delta _{0})>\mu ^{*}_{2n+1,2m_{n}}(\Delta _{1})$ . Therefore,

$$ \begin{align*} \frac{\nu^{*}_{2n+1,2m_{n}}(\Delta_{0})}{\nu^{*}_{2n+1,2m_{n}}(\Delta_{1})}= \frac{\unicode{x3bb}\mu^{*}_{2n+1,2m_{n}}(\Delta_{0})}{\mu^{*}_{2n+1,2m_{n}}(\Delta_{1})}\ge \unicode{x3bb}. \end{align*} $$

Since the action $\bigoplus _{j=2n+1}^{2m_{n}} S^{*}$ of $G^{2(m_{n}-n)}$ on $((Y^{*})^{2(m_{n}-n)},\nu ^{*}_{2n+1,2m_{n}})$ is measure preserving and ergodic, there exist a subset $\Delta ^{\prime }_{1}\subset \Delta _{1}$ and a transformation Q from the full group $[\bigoplus _{j=2n+1}^{2m_{n}} S^{*},\nu ^{*}_{2n+1,2m_{n}}]$ such that $Q\Delta ^{\prime }_{1}\subset \Delta _{0}$ and $\nu ^{*}_{2n+1,2m_{n}}(\Delta ^{\prime }_{1})=\unicode{x3bb} \nu ^{*}_{2n+1,2m_{n}}(\Delta _{1})$ . Then,

$$ \begin{align*} \mu^{*}_{2n+1,2m_{n}}(\Delta_{1}^{\prime})=\unicode{x3bb}\nu^{*}_{2n+1,2m_{n}}(\Delta_{1}^{\prime})=\unicode{x3bb}^{2}\nu^{*}_{2n+1,2m_{n}}(\Delta_{1})=\unicode{x3bb}\mu^{*}_{2n+1,2m_{n}}(\Delta_{1})>\frac{\unicode{x3bb}}{16}. \end{align*} $$

If $\omega \in \Delta _{1}^{\prime }$ , then $Q\omega \in \Delta _{0}$ and hence $({d\mu ^{*}_{2n+1,2m_{n}}}/{d\nu ^{*}_{2n+1,2m_{n}}})(\kern-1pt Q\omega )\kern-1pt =\kern-1pt 1$ and $({d\mu ^{*}_{2n+1,2m_{n}}} / {d\nu ^{*}_{2n+1,2m_{n}}})(\omega )=\unicode{x3bb} $ by the definition of $\Delta _{0}$ and $\Delta _{1}$ . Therefore,

$$ \begin{align*} \frac{d\mu^{*}_{2n+1,2m_{n}}\circ Q}{d\mu^{*}_{2n+1,2m_{n}}}(\omega)= \frac{d\mu^{*}_{2n+1,2m_{n}}}{d\nu^{*}_{2n+1,2m_{n}}}(Q\omega)\frac{d\nu^{*}_{2n+1,2m_{n}}}{d\mu^{*}_{2n+1,2m_{n}}}(\omega)=\unicode{x3bb}^{-1}. \end{align*} $$

Now, take a Borel subset C of $(Y^{*})^{2n}$ . Then:

• • $[C\times \Delta ^{\prime }_{1}]_{2m_{n}}\subset [C]_{2n}$ ;

• • $\mu ^{*}([C\times \Delta ^{\prime }_{1}]_{2m_{n}})=\mu ^{*}([C]_{2n})\mu ^{*}_{2n+1,2m_{n}}(\Delta _{1}^{\prime })>(\unicode{x3bb} /{16})\mu ^{*}([C]_{2n})$ ; and

• • $I\times Q\times I\in [\bigoplus _{n=1}^{\infty } S^{*},\mu ^{*}]$ and $ (({d\mu ^{*}\circ (I\times Q\times I)})/{d\mu ^{*}})(\omega )=\unicode{x3bb} ^{-1} $ for $\mu ^{*}$ -a.e. $\omega \in [C\times \Delta ^{\prime }_{1}]_{2m_{n}}$ .

Since the family $\{[C]_{2n}\mid C\subset (Y^{*})^{2n}, n\in \Bbb N\}$ is dense in the entire Borel $\sigma $ -algebra on $X^{*}$ , it follows that $\unicode{x3bb} ^{-1}\in r(\bigoplus _{n=1}^{\infty } S^{*},\mu ^{*})$ by Fact 2.7. On the other hand, the Radon–Nikodym cocycle of the system $(X^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ takes its values in the subgroup $\{\unicode{x3bb} ^{n}\mid n\in \Bbb Z\}$ of $\Bbb R_{+}^{*}$ . It follows that $r(\bigoplus _{n=1}^{\infty } S^{*},\mu ^{*})\subset \{\unicode{x3bb} ^{n}\mid n\in \Bbb Z\}$ . Hence, $r(\bigoplus _{n=1}^{\infty } S^{*},\mu ^{*})=\{\unicode{x3bb} ^{n}\mid n\in ~\Bbb Z\}$ . Therefore, $(X^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ is of type $III_{\unicode{x3bb} }$ , as desired. Thus, the claim is proved.

It follows from Claim A and Proposition 2.17 that $(X^{*},\mu ^{*},T^{*})$ is of type $III_{\unicode{x3bb} }$ .

Remark 6.2. Arguing as in Remark 5.2, one can choose the parameters $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ in such a way that the Poisson suspension of the corresponding action T (we mean the general construction of §3) is indeed of infinite ergodic index. We leave the details to the reader.

Theorem 6.3. If G is Haagerup, then there is a dynamical system $(X,\mathfrak {B},\mu , T)$ such that the Poisson suspension $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ of $(X,\mathfrak {B},\mu , T)$ is of 0-type (in addition to the other properties of $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ listed in Theorem 6.1).

Proof. Since G is Haagerup, there is a free 0-type G-action $S=(S_{g})_{g\in G}$ on an infinite $\sigma $ -finite standard measure space $(Y,\mathfrak {Y},\kappa )$ such that $(\alpha _{1})$ – $(\alpha _{5})$ , (5.1) and $(\alpha _{6})^{\prime }$ hold.

We fix an increasing sequence $H_{1}\subset H_{2}\subset \cdots $ of compact subsets in G with $\bigcup _{n=1}^{\infty } H_{n}=G$ . Then, there is a sequence $K_{1}\subset K_{2}\subset \cdots $ of compact subsets in G with $\bigcup _{n=1}^{\infty } K_{n}=G$ and a sequence of positive reals $n_{1}<n_{2}<\cdots $ such that, for each $l\in \Bbb N$ ,

1. (α ₇) $\sup _{g\in K_{l}}\kappa (S_{g}F_{n_{l}}\triangle F_{n_{l}})<1/{l^{2}}$ and

2. (α ₈) $\sup _{1\le j\le l}\sup _{g\not \in K_{l+1}}(1-\kappa (S_{g}F_{n_{j}}\cap F_{n_{j}}))<1/{l}$ .

These sequences can be constructed inductively. If we have already determined $(n_{j})_{j=1}^{l-1}$ and $(K_{j})_{j=1}^{l}$ , then there exists $n_{l}$ such that $(\alpha _{7})$ holds. This follows from the fact that $(F_{n})_{n=1}^{\infty }$ is S-Følner and $\kappa (F_{n})=1$ for each $n\in \Bbb N$ . Then, we select a compact subset $K_{l+1}\subset G$ large enough so that $K_{l+1}\supset K_{l}\cup H_{l}$ and $(\alpha _{8})$ holds. This follows from the fact that S is of 0-type and $\kappa (F_{n})=1$ for each $n\in \Bbb N$ . By repeating these two steps infinitely many times, we determine the entire sequences $(n_{j})_{j=1}^{\infty }$ and $(K_{j})_{j=1}^{\infty }$ .

To simplify the notation, we rename the subsequence $(F_{n_{l}})_{l=1}^{\infty }$ into $(F_{n})_{n=1}^{\infty }$ . Thus, from now on, we may assume, without loss of generality, that $(\alpha _{1})$ – $(\alpha _{5})$ , (5.1) and $(\alpha _{6})^{\prime }$ hold and, in addition, for each $n\in \Bbb N$ ,

1. (α ₇)^′ $\sup _{g\in K_{n}}\kappa (S_{g}F_{n}\triangle F_{n})<1/{n^{2}}$ and

2. (α ₈)^′ $\sup _{1\le j\le n}\sup _{g\not \in K_{n+1}}(1-\kappa (S_{g}F_{j}\cap F_{j}))<1/n$ .

Repeating the proof of Theorem 6.1 almost literally, we construct a system $(X,\mathfrak {B},\mu ,T)$ such that the Poisson suspension $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ of $(X,\mathfrak {B},\mu ,T)$ is well defined and possesses all the properties listed in Theorem 6.1. It remains to prove that $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ is of 0-type. For that, we first show that

(6.4) $$ \begin{align} \lim_{g\to\infty} \sum_{n=1}^{\infty} \frac 1{n\log(n+1)}\kappa(S_{g}F_{n}\triangle F_{n})= 0. \end{align} $$

Given $\epsilon>0$ , we select $N>0$ such that $\sum _{n>N}( 1/ {n^{2}})<\epsilon $ . Let $g\not \in K_{N}$ . Then, there is $l\ge N$ such that $g\in K_{l+1}\setminus K_{l}$ . It follows from $(\alpha _{7})^{\prime }$ that

$$ \begin{align*} \sum_{n>l} \frac 1{n\log(n+1)}\kappa(S_{g}F_{n}\triangle F_{n})<\sum_{n>l} \frac 1{n^{2}}<\epsilon. \end{align*} $$

On the other hand, it follows from $(\alpha _{8})^{\prime }$ that

$$ \begin{align*} \sum_{n=1}^{l} \frac 1{n\log(n+1)}\kappa(S_{g}F_{n}\triangle F_{n})< \frac1{l-1}\sum_{n=1}^{l-1}\frac 1{n\log(n+1)} + \frac{2}{l\log(l+1)}. \end{align*} $$

Since the right-hand side of this inequality is less than $\epsilon $ whenever N is large enough, (6.4) follows. We now deduce from [DaKoRo1, Proposition 6.9], $(a_{6})^{\prime }$ and (6.4) that $T^{*}$ is of zero type.

7 Type $III_{1}$ ergodic Poisson suspensions

In this section, we prove the implications (1) $\Rightarrow $ (3) of Theorems A and B for $K=III_{1}$ . Thus, we assume that G does not have property (T). Fix $\unicode{x3bb} _{1},\unicode{x3bb} _{2}\in (0,1)$ such that $\log \unicode{x3bb} _{1}$ and $\log \unicode{x3bb} _{2}$ are rationally independent. The following theorem follows from Theorem 6.1 (as above, we use here the notation from §3). For $n\in \Bbb N$ , let

$$ \begin{align*} \unicode{x3bb}_{4n-3}&:=\unicode{x3bb}_{4n-2}^{-1}:=\unicode{x3bb}_{1}, \\ \unicode{x3bb}_{4n-1}&:=\unicode{x3bb}_{4n}^{-1}:=\unicode{x3bb}_{2},\\ a_{4n-3}&:=\unicode{x3bb}^{-1}_{1}a_{4n-2}:=\frac 1{n\log (n+1)} \quad\text{and }\\ a_{4n-1}&:=\unicode{x3bb}_{2}^{-1}a_{4n}:=\frac 1{n\log (n+1)}. \end{align*} $$

Denote by $(X,\mathfrak {B},\mu , T)$ the dynamical system associated with $(Y,\mathfrak {Y},\kappa ,S)$ , $\boldsymbol {F}$ , $(a_{n})_{n=1}^{\infty }$ and $(\unicode{x3bb} _{n})_{n=1}^{\infty }$ via the general construction from §3.

Theorem 7.1. The Poisson suspension $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ of $(X,\mathfrak {B},\mu , T)$ is a well defined non-singular dynamical system. It is free, non-strongly ergodic, IDPFT, of Krieger type $III_{1}$ and of infinite ergodic index. Hence, $T^{*}$ is weakly mixing and amenable in the Greenleaf sense. Moreover, $T^{*}$ is amenable if and only if G is amenable.

Idea of the proof.

Follow the proof of Theorem 6.1 to show that the associated flow of $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is isomorphic to the associated flow of the the direct sum $(X^{*},\mathfrak {B}^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ by Proposition 2.17. However, the latter system splits into the direct sum (do not confuse with the direct product) of the following two systems: $(\bigotimes _{j\in I} (Y^{*},\mu _{j}^{*}),\bigoplus _{j\in I} S^{*})$ and $(\bigotimes _{j\in J} (Y^{*},\mu _{j}^{*}),\bigoplus _{j\in J} S^{*})$ , where

$$ \begin{align*} I&:=\{4n-2\mid n\in\Bbb N\}\cup\{4n-3\mid n\in\Bbb N\}\quad\text{and }\\ J&:=\{4n-1\mid n\in\Bbb N\}\cup\{4n\mid n\in\Bbb N\}. \end{align*} $$

These two dynamical systems are of type $III_{\unicode{x3bb} _{1}}$ and $III_{\unicode{x3bb} _{2}}$ , respectively, according to Claim A from the proof of Theorem 6.1. Since $\log \unicode{x3bb} _{1}$ and $\log \unicode{x3bb} _{2}$ are rationally independent, the associated flow of $(X^{*},\mathfrak {B}^{*},\mu ^{*},\bigoplus _{n=1}^{\infty } S^{*})$ is trivial (acting on a singleton). Hence, $(X^{*},\mathfrak {B}^{*},\mu ^{*},T^{*})$ is of type $III_{1}$ . The other properties of this system can be established in the same way as in the proof of Theorem 6.1.

In a similar way, one can prove the following assertion.

Theorem 7.2. If G is Haagerup, then there is a dynamical system $(X,\mathfrak {B},\mu , T)$ such that the Poisson suspension $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ of $(X,\mathfrak {B},\mu , T)$ is a well defined non-singular dynamical system. This system is of 0-type and possesses all the properties of $( X^{*},\mathfrak {B}^{*},\mu ^{*}, T^{*})$ listed in Theorem 7.1.

8 Applications to non-singular Bernoulli actions of locally compact amenable groups

Let G be a non-compact locally compact second countable group. Given an infinite $\sigma $ -finite standard measure space $(X,\mathfrak {B},\mu )$ , we let (as in [DaKoRo1])

$$ \begin{align*} \textrm{{Aut}}_{2}(X,\mu):=\bigg\{S\in\textrm{{Aut}}(X,\mu)\,\bigg|\,\sqrt{\frac{d\mu\circ S}{d\mu}}-1\in L^{2}(X,\mu)\bigg\}. \end{align*} $$

As was shown in [DaKoRo1], given a transformation $S\in \textrm {{Aut}}(X,\mu )$ , the non-singular Poisson suspension $S^{*}$ of S is well defined (as an element of $\textrm {{Aut}}(X^{*},\mu ^{*})$ ) if and only if $S\in \textrm {{Aut}}_{2}(X,\mu )$ . In particular, Aut $_{1}(X,\mu )\subset \text {Aut}_{2}(X,\mu )$ .

Definition 8.1. Let $T=(T_{g})_{g\in G}$ be a free totally dissipative non-singular G-action on a $\sigma $ -finite standard measure space $(X,\mathfrak {B},\mu )$ such that $T_{g}\in \textrm {{Aut}}_{2}(X,\mu )$ for each $g\in G$ . Then, we call the Poisson suspension $T^{*}:=(T^{*}_{g})_{g\in G}$ of T the non-singular Bernoulli G-action over the base $(X,\mathfrak {B},\mu ,T)$ .

If G is discrete countable, then each non-singular Bernoulli G-action according to Definition 8.1 is non-singular Bernoulli in the usual sense. However, we do not know whether each non-singular Bernoulli G-action in the usual sense is isomorphic to a Bernoulli G-action in the sense of Definition 8.1.

For an arbitrary G, if T preserves $\mu $ , then $T^{*}$ is a probability-preserving Bernoulli G-action in the sense of [OrWe].

Let $\unicode{x3bb} _{G}$ denote a left Haar measure on G. Since T is totally dissipative and free, there is a Borel subset $B\subset X$ which meets a.e. T-orbit exactly once. Hence, there exist a $\sigma $ -finite Borel measure $\kappa $ on B and a Borel isomorphism (mod 0) of X onto $B\times G$ such that T corresponds to G-action by the left translations along the second coordinate of $B\times G$ . We identify X and $B\times G$ via this isomorphism. Choose a $\sigma $ -finite Borel measure $\kappa $ on B which has the same collection of subsets of zero measure as the projection of $\mu $ to B. Then, the disintegration $\mu =\int _{B}\delta _{b}\otimes \mu _{b}\,d\kappa (b)$ of $\mu $ with respect to $\kappa $ is well defined. Of course, $\mu _{b}\sim \unicode{x3bb} _{G}$ at a.e. $b\in B$ . Therefore, there is a measurable function $F:B\times G\to \Bbb R^{*}_{+}$ such that $F(b,g)=({d\mu _{b}}/{d\unicode{x3bb} _{G}})(g)$ almost everywhere. The condition $T_{h}\in \textrm {{Aut}}_{2}(X,\mu )$ can be rewritten now as

$$ \begin{align*} \int_{B}\int_{G}(\sqrt{F(b,g)}-\sqrt{F(b,hg)})^{2}\,d\unicode{x3bb}_{G}(g)d\kappa(b)<\infty \end{align*} $$

for each $h\in G$ . In particular, letting $c_{b,h}(g):=\sqrt {F(b,g)}-\sqrt {F(b,hg)}$ , we obtain that $c_{b,h}\in L^{2}(G,\unicode{x3bb} _{G})$ for each $b\in B$ and $h\in H$ . Moreover, the function $B\ni b\mapsto \|c_{b,h}\|_{2}$ belongs to $L^{2}(B,\kappa )$ .

Definition 8.2. If there is a measurable function $d:B\ni b\mapsto d_{b}\in L^{2}(G,\unicode{x3bb} _{G})$ such that $c_{b,h}(g)=d_{b}(g)-d_{b}(hg)$ at a.e. $g\in G$ and each $h\in H$ , then we say that $(X,\mathfrak {B},\mu , T)$ is tame.

If $(X,\mathfrak {B},\mu , T)$ is tame, then we can find a countable partition $B=\bigsqcup _{j}B_{j}$ of B into measurable subsets such that the function $B_{j}\ni b\mapsto \|d_{b}\|_{2}$ belongs to $L^{2}(B_{j},\kappa \restriction B_{j})$ for each j. Then, we get a countable partition $\bigsqcup _{j}(B_{j}\times G)$ of X into T-invariant subsets furnished with T-invariant measures $\nu _{j}$ such that $\sqrt {{d\mu }/{d\nu _{j}}}-1\in L^{2}(B_{j}\times G,\nu _{j})$ for each j. It follows that $((B_{j}\times G)^{*},(\mu \restriction (B_{j}\times G))^{*}, T^{*})$ is non-singular Bernoulli of type $II_{1}$ . Thus, if $(X,\mathfrak {B},\mu , T)$ is tame, then the corresponding Bernoulli G-action over $(X,\mathfrak {B},\mu , T)$ is IDPFT.

Corollary 8.3. Let G be amenable non-compact locally compact second countable group. For each $K\in \{III_{\unicode{x3bb} }\mid 0\le \unicode{x3bb} \le 1\}\cup \{II_{\infty }\}$ , there is a tame Bernoulli free non-singular G-action T of infinite ergodic index and of Krieger type K. Hence, T is weakly mixing, IDPFT and amenable in the Greenleaf sense.

Proof. Since G is amenable, there is a left Følner sequence $(F_{n})_{n=1}^{\infty }$ on G. Let $(Y,\kappa ):=(G,\unicode{x3bb} _{G})$ . Let S denote the G-action on Y by left translations and let $\boldsymbol {F}:=(F_{n})_{n=1}^{\infty }$ . Then, the Koopman representation $U_{S}$ is the left regular representation of G. Hence, $U_{S}$ is mixing, that is, S is of 0-type. Moreover, $\boldsymbol {F}$ is S-Følner. It remains to apply the constructions from Theorems 4.4, 5.3, 6.3 and 7.2 (depending on the value of K) to the quadruple $(Y,\kappa ,S,\boldsymbol {F})$ .

9 Interplay between non-singular Poisson and non-singular Gaussian actions

Given a real separable Hilbert space $\mathcal {H}$ , we denote by $\mathcal {A}$ the group of affine transformations of $\mathcal {H}$ , that is, $\mathcal {A}=\mathcal {H}\rtimes \mathcal {O}$ , where $\mathcal {O}$ is the group of orthogonal transformations of $\mathcal {H}$ . A transformation $(h,O)\in \mathcal {A}$ acts on a vector $v\in \mathcal {H}$ by the formula $(h,O)v:=h+Ov$ . It was shown in [ArIsMa] (see also [DaLe, §3]) that, given a continuous homomorphism $\alpha :G\to \mathcal {A}$ , one can construct a non-singular G-action, called the non-singular Gaussian action generated by $\alpha $ .

Now, suppose that we have an infinite $\sigma $ -finite standard measure space $(Y,\mathfrak {Y},\nu )$ . Let $S=(S_{g})_{g\in G}$ be a non-singular G-action on Y such that $S_{g}\in \text {Aut}_{2}(Y,\nu ) $ for all $g\in G$ . Then, the Poisson suspension $S^{*}=(S_{g}^{*})_{g\in G}$ is well defined as a non-singular G-action on $(X^{*},\mathfrak {Y}^{*},\nu ^{*})$ [DaKoRo1]. Denote by $\mathcal {A}$ the group of affine transformations of $L^{2}(Y,\nu )$ . Then, a continuous homomorphism

$$ \begin{align*} A_{S}:G\ni g\mapsto A_{S}(g)\in \mathcal{A} \end{align*} $$

is well defined by the formula $ A_{S}(g):=(2(\sqrt {({d\nu \circ S_{g}^{-1}})/{d\nu }} -1 ),U_{S}(g))\in \mathcal {A}$ . We call $A_{S}$ the affine Koopman representation of G associated with S [DaKoRo1]. Denote by $H=(H_{g})_{g\in G}$ the non-singular Gaussian G-action generated by $A_{S}$ . It was explained in [DaLe, Remark 3.6] that H and $S^{*}$ are spectrally identical, that is, the unitary Koopman representations $U_{H}$ and $U_{S^{*}}$ of G are unitarily equivalent. Hence, it is of interest to compare non-spectral dynamical properties of H and $S^{*}$ .

Proposition 9.1. Let G have the Haagerup property. Let S be a 0-type non-singular G-action on the standard $\sigma $ -finite measure space $(Y,\mathfrak {Y},\nu )$ such that $S_{g}\in \textrm {{Aut}}_{2}(Y,\nu )$ for each $g\in G$ . Suppose that $Y=\bigsqcup _{n=1}^{\infty } Y_{n}$ for some S-invariant subsets $Y_{n}\in \mathfrak {Y}$ of infinite measure $\nu $ and there is an S-invariant $(\nu \restriction Y_{n})$ -equivalent $\sigma $ -finite measure $\xi _{n}$ on $Y_{n}$ with $\sqrt {{d\xi _{n}}/{d\nu }}-1\in L^{2}(Y_{n},\nu )$ for each $n\in \Bbb N$ . If the Poisson suspension $(Y^{*},\nu ^{*},S^{*})$ does not admit an invariant equivalent probability measure, then H is either totally dissipative or weakly mixing of type $III_{1}$ . In particular, if G is non-amenable, then H is weakly mixing of type $III_{1}$ .

Proof. First, we note that $L^{2}(Y,\nu )=\bigoplus _{n=1}^{\infty } L^{2}(Y_{n},\nu )$ and that $L^{2}(Y_{n},\nu )$ is an invariant subspace for $U_{S}$ for each $n\in \Bbb N$ . Let

$$ \begin{align*} c_{g}:=2\bigg(\sqrt{\frac{d\nu\circ S_{g}^{-1}}{d\nu}} -1\bigg)\in L^{2}(Y,\nu)\quad\text{for each }g\in G\text{.} \end{align*} $$

Consider $ L^{2}(Y,\nu )$ as a G-module, where G acts via $U_{S}$ . Then, the mapping

$$ \begin{align*} c:G\ni g\mapsto c_{g}\in L^{2}(Y,\nu) \end{align*} $$

is a 1-cocycle of G with coefficients in $L^{2}(Y,\nu )$ . It follows from the condition of the proposition and [DaKoRo1, Proposition 6.4] that, for each $n>0$ , the cocycle $G\ni g\mapsto 1_{Y_{n}}c_{g}\in L^{2}(Y_{n},\nu )$ is a coboundary. Hence, there exists a function $a_{n}\in L^{2}(Y_{n},\nu )$ such that $1_{Y_{n}}c_{g}= a_{n}-U_{S}(g)a_{n}$ for each $g\in G$ . We now set

$$ \begin{align*} \mathcal{H}_{n}^{0}:=\bigoplus_{m>n}L^{2}(Y_{m},\nu)\quad\text{and}\quad \mathcal{H}_{n}:=a_{1}\oplus\cdots \oplus a_{n}\oplus\mathcal{H}_{n}^{0}. \end{align*} $$

Then, $\mathcal {H}_{1}^{0}\supset \mathcal {H}_{2}^{0}\supset \cdots $ , $\bigcap _{n=1}^{\infty }\mathcal {H}^{0}=\{0\}$ and

$$ \begin{align*} A_{S}(g)\mathcal{H}_{n}=\bigoplus_{m=1}^{n}(U_{S}(g)a_{m}+1_{Y_{m}}c_{g})\oplus U_{S}(g)\mathcal{H}^{0}_{n}=\bigoplus_{m=1}^{n} a_{n}\oplus\mathcal{H}_{n}^{0}=\mathcal{H}_{n} \end{align*} $$

for each $g\in G$ and $n\in \Bbb N$ . Thus, $\mathcal {H}_{n}$ is invariant under $A_{S}$ . Hence, by [ArIsMa, Proposition 2.10], $A_{S}$ is evanescent according to [ArIsMa, Definition 2.6]. Since S is of 0-type, $U_{S}$ is mixing. Since $(Y^{*},\nu ^{*},S^{*})$ does not admit an invariant equivalent probability measure, then c is not a coboundary [DaKoRo1, Proposition 6.4]. Therefore, [ArIsMa, Theorem D] yields that H is either totally dissipative or weakly mixing and of Krieger type $III_{1}$ . The first claim of the proposition is proved.

Now, suppose that G is non-amenable. We have to show that H is not totally dissipative. It follows from the condition of the proposition that $(Y^{*},\nu ^{*},S^{*})$ is IDPFT. Hence, by Proposition 2.25, $S^{*}$ is amenable in the Greenleaf sense. Since the amenability in the Greenleaf sense is an invariant for the unitary equivalence, it follows that H is also amenable in the Greenleaf sense. We deduce from this fact (as G is non-amenable) that H is not amenable in view of Fact 2.24 (ii), (iv) and (v). Since each totally dissipative G-action is amenable, H is not totally dissipative.

Corollary 9.2. Let G be non-amenable and have the Haagerup property. Then, for all non-singular Poisson G-actions (of any Krieger type) constructed in Theorem B, the corresponding non-singular Gaussian G-actions are all weakly mixing and of Krieger type $III_{1}$ .