Bernoulli actions of type III 0 with prescribed associated ﬂow

We prove that many, but not all injective factors arise as crossed products by nonsingular Bernoulli actions of the group Z . We obtain this result by proving a completely general result on the ergodicity, type and Krieger’s associated ﬂow for Bernoulli shifts with arbitrary base spaces. We prove that the associated ﬂow must satisfy a structural property of inﬁnite divisibility. Conversely, we prove that all almost periodic ﬂows, as well as many other ergodic ﬂows, do arise as associated ﬂow of a weakly mixing Bernoulli action of any inﬁnite amenable group. As a byproduct, we prove that all injective factors with almost periodic ﬂow of weights are inﬁnite tensor products of 2 × 2 matrices. Finally, we construct Poisson suspension actions with prescribed associated ﬂow for any locally compact second countable group that does not have property (T).


Introduction
To a countable infinite group G and a standard measure space (X 0 , µ 0 ), called the base space, one associates the Bernoulli action G (X, µ G 0 ) = g∈G (X 0 , µ 0 ) given by translating the coordinates by left multiplication.Bernoulli actions are at the heart of many classical, as well as recent theorems in ergodic theory and operator algebras.Especially the role of Bernoulli actions in the theory of von Neumann algebras has been very prominent, see [Pop03,Pop06,CI09,Ioa10,PV21].
By construction µ G 0 is a probability measure and it is preserved by the Bernoulli action of G.We rather equip X = X G 0 with a product of possibly distinct probability measures µ g on X 0 and thus consider the Bernoulli action G (X, µ) = g∈G (X 0 , µ g ) : (g −1 • x) h = x gh . (1.1) We require that the action (1.1) is nonsingular, i.e. preserves sets of measure zero.By Kakutani's criterion for the equivalence of product measures, this is equivalent to all the measures (µ g ) g∈G being equivalent and h∈G H 2 (µ gh , µ h ) < +∞ for every g ∈ G, (1.2) where H(µ, ν) denotes the Hellinger distance, see (2.1).
The key question that we address is the following: given a countable infinite group G, what are the possible Krieger types of nonsingular Bernoulli actions G (X, µ) ?This question is particularly interesting in the classical case G = Z.
Recall that an essentially free ergodic nonsingular action G (X, µ) is said to be of type II 1 if it admits an equivalent G-invariant probability measure, of type II ∞ if it admits an equivalent G-invariant infinite measure and of type III otherwise.Moreover, type III actions are further classified by Krieger's associated flow [Kri76], an ergodic nonsingular action of R that is also equal to the Connes-Takesaki flow of weights [CT77] of the crossed product von Neumann algebra L ∞ (X) ⋊ G.If the associated flow is trivial, the action is of type III 1 .If it is periodic with period | log λ| and λ ∈ (0, 1), the action is of type III λ .If the associated flow is properly ergodic, the action is of type III 0 and we are particularly interested to understand which associated flows may arise from nonsingular Bernoulli actions.
The first example of an ergodic Bernoulli action of type III was given by Hamachi for the group of integers [Ham81].Much later in [Kos09], Kosloff could give an example of a nonsingular Bernoulli action of Z that is of type III 1 .
In the past few years, the study of nonsingular Bernoulli actions has gained momentum.The first systematic results for nonsingular Bernoulli actions of nonamenable groups G were obtained in [VW17].In [BKV19], very complete results on the ergodicity and type of nonsingular Bernoulli actions with base space X 0 = {0, 1} were obtained, building on important earlier work in [Kos12,DL16,Kos18,Dan18].In particular, it was shown in [VW17] that the free groups F n , n ≥ 2, admit Bernoulli actions of type III λ for all λ ∈ (0, 1].In [BKV19], it was proven that locally finite groups admit Bernoulli actions of all possible types: II 1 , II ∞ and III λ , for λ ∈ [0, 1].In [BV20], we proved that the same holds for all infinite amenable groups, if we allow the base space X 0 to be infinite.The latter is a necessary assumption, since it was proven in [BKV19] that Bernoulli actions of Z with finite base space are never of type II ∞ .In [KS20], it was proven independently that infinite amenable groups admit Bernoulli actions of type III λ for all λ ∈ (0, 1].Ergodic, essentially free, nonsingular actions G (X, µ) of amenable groups are completely classified, both up to orbit equivalence and up to isomorphism of their crossed product von Neumann algebras, by their type and associated flow, see [Kri76,Con76,CT77,CFW81,Haa85].It is thus a very natural question to ask which ergodic flows arise as the associated flow of a nonsingular Bernoulli action, in particular of the group Z.Put in an equivalent form, the question is which factors arise as crossed products L ∞ (X) ⋊ Z by nonsingular Bernoulli shifts.
We prove in this paper the surprising result that not all injective factors can arise in this way.We also prove that many injective type III 0 factors do arise.In particular, we prove that all infinite tensor products of factors of type I 2 (i.e. 2 × 2 matrices), the so called ITPFI 2 factors, arise as crossed products of nonsingular Bernoulli shifts.
In this paper, we call a flow any nonsingular action of R. We introduce below (see Definition 3.12) the concept of an infinitely divisible flow.By the classification of injective factors, the flow of weights of an injective factor M is infinitely divisible if and only if for every integer n ≥ 1, there exists an injective factor N such that M ∼ = N ⊗n .By [GSW84, Theorem 2.1], not every injective factor, and not even every ITPFI factor, is a tensor square.So not all ergodic flows are infinitely divisible.
Our first main result says that the associated flow of a nonsingular Bernoulli shift Z (X, µ) must be infinitely divisible.We prove this result in complete generality, without making any other assumptions on the nature of the base space X 0 or the probability measures µ n , apart from the shift being nonsingular.
We thus also need a completely general result on the ergodicity of nonsingular Bernoulli shifts.Ruling out the trivial cases where µ admits an atom or where the action Z (X, µ) admits a fundamental domain (i.e. is dissipative), we prove the following result.We actually provide in Theorem 4.1 below a more precise description, also saying exactly what happens in the trivial cases with an atom or a fundamental domain.
There exists an essentially unique Borel set C 0 ⊂ X 0 such that C Z 0 ⊂ X has positive measure and the following holds.
• The nonsingular Bernoulli shift Z C Z 0 is weakly mixing and its associated flow is infinitely divisible.
• The action Z X \ C Z 0 is dissipative.
Note that it was proven in [BKV19, Theorem A] that a Bernoulli shift of Z with base space {0, 1} is either weakly mixing, dissipative or atomic.This is compatible with Theorem A because a two point base space is the only case in which a subset C 0 ⊂ X 0 is either empty, a single point or everything.Theorem A says in particular that for every conservative nonsingular Bernoulli shift Z (X, µ), the associated flow is infinitely divisible.The crossed product L ∞ (X) ⋊ Z associated with a Bernoulli shift can only be a factor if Z (X, µ) is conservative and ergodic.As mentioned above, by [GSW84, Theorem 2.1], it thus follows that not every injective factor, and not even every ITPFI factor, is of the form L ∞ (X) ⋊ Z where Z (X, µ) is a nonsingular Bernoulli shift.
Complementing Theorem A, we determine in Theorem 4.3 in equally complete generality the type of an arbitrary nonsingular Bernoulli shift Z (X, µ).
In the converse direction, we prove that many ergodic flows do arise as associated flows of nonsingular Bernoulli actions.By [CW88], the possible flows of weights of ITPFI factors are precisely the tail boundary flows, i.e. the actions of R on the Poisson boundary of a time dependent random walk on R given by a (nonconstant) sequence of transition probability measures µ n on R. If the transition probabilities µ n can be chosen to be compound Poisson distributions, we call the tail boundary flow a Poisson flow4 , see Definition 3.7.
We prove in Theorems 3.8 and 3.9 that the class of Poisson flows is large: it includes all almost periodic flows (i.e.flows with pure point spectrum) and it includes the flow of weights of any ITPFI 2 factor.By definition, Poisson flows are infinitely divisible and therefore, not every ergodic flow is a Poisson flow.In Section 3, we obtain several results on the class of Poisson flows.They can be equivalently characterized as the tail boundary flows with transition probabilities µ n supported on two points and having uniformly bounded variance (see Proposition 3.11).Also, the flows of weights of ITPFI 2 factors can be precisely characterized as the Poisson flows of positive type (see Definition 3.7 and Theorem 3.9).
We then prove that all these Poisson flows arise as the associated flow of a weakly mixing nonsingular Bernoulli action of any amenable group.As a corollary, it thus follows that every ITPFI 2 factor is of the form L ∞ (X) ⋊ Z for a nonsingular, weakly mixing Bernoulli shift Z (X, µ).
Theorem B. Let G be any countable infinite amenable group and let R (Z, η) be any Poisson flow.There exists a family of equivalent probability measures (µ g ) g∈G on a countable infinite base space X 0 such that the Bernoulli action G (X, µ) = g∈G (X 0 , µ g ) is nonsingular, weakly mixing and has associated flow R Z.
As a byproduct of our results on Poisson flows, it follows that every almost periodic flow is the flow of weights of an ITPFI 2 factor, answering a question that remained open since [HO83,GS84].
Theorem C. Every injective factor with almost periodic flow of weights is an ITPFI 2 factor.
To put Theorem C in a proper context, recall that an ergodic almost periodic flow is precisely given by the translation action R Λ where Λ ⊂ R is a countable subgroup.So, for every countable subgroup Λ ⊂ R, there is a unique injective factor M Λ with flow of weights the translation action R Λ. Connes' T -invariant of M Λ equals Λ.In [HO83, Theorem 2], it was proven that for every α ∈ R \ {0} and every subgroup Λ ⊂ αQ, the factor M Λ is ITPFI 2 .In [GS84, Proposition 1.1], it was proven that for every countable subgroup Λ ⊂ R, there exists an ITPFI 2 factor M with T (M ) = Λ, but it remained unclear if M ∼ = M Λ .We now prove in Theorem C that all M Λ are ITPFI 2 factors.
Every ITPFI 2 factor is infinitely divisible.It is natural to speculate that also the converse holds.One may also speculate that the infinitely divisible ergodic flows are exactly the Poisson flows.If both of these speculations are true, it follows from Theorem A and Theorem B that the class of injective factors that can be realized as the crossed product L ∞ (X) ⋊ Z by a conservative nonsingular Bernoulli action Z (X, µ) equals the class of ITPFI 2 .We refer to Remark 3.16 for a further discussion.
Poisson flows also appear naturally in the context of nonsingular Poisson suspensions (see Section 5 for terminology).Given an infinite, σ-finite, standard measure space (X 0 , µ 0 ) with Poisson suspension (X, µ), under the appropriate assumptions, a nonsingular action G (X 0 , µ 0 ) has a canonical nonsingular suspension G (X, µ).To a certain extent, Poisson suspensions can be viewed as generalizations of Bernoulli actions, and many results were obtained recently (see [Roy08,DK20,DKR20,Dan21]). In particular in [Dan21], it was proven that any locally compact second countable group G that does not have property (T) admits nonsingular Poisson suspension actions of any possible type.We prove that in type III 0 , any Poisson flow may arise as associated flow.
Proposition D. Let G be any locally compact second countable group that does not have property (T).Let R (Z, η) be any Poisson flow.Then G admits a nonsingular action G (X 0 , µ 0 ) of which the Poisson suspension G (X, µ) is well-defined, weakly mixing, essentially free and has associated flow R Z.
By [DKR20, Theorem G], Proposition D is sharp: if G has property (T), then every nonsingular Poisson suspension action of G admits an equivalent G-invariant probability measure.We actually prove first Proposition D and then deduce Theorem B as a special case, by taking G G × N : g • (h, n) = (gh, n) and choosing the measure µ 0 on G × N appropriately.
Acknowledgment.We thank Alexandre Danilenko for his useful comments on the first draft of this paper.
We say that ϕ is nonsingular if the measures ϕ * µ and µ are equivalent.If in addition ϕ is invertible, we say it is a nonsingular automorphism.In that case we will also use the notation µ • ϕ for the push forward measure ϕ −1 µ.The set Aut(X, µ) is the group of all nonsingular automorphisms of (X, µ), where we identify two elements if they agree almost everywhere.It caries a canonical topology, making it into a Polish group.Both the set Aut(X, µ) and its topology only depend on the measure class of µ.
For a nonsingular automorphism ϕ : X → X the Radon-Nikodym derivative is uniquely determined by the equality A nonsingular action of a locally compact second countable group H on a standard measure space (X, µ) is a continuous homomorphism α : H → Aut(X, µ), which we will also write as We call a flow any nonsingular action of A nonsingular action G (X, µ) of a countable group G on a standard measure space (X, µ) is called essentially free if the set {x ∈ X : g • x = x} has measure zero for every g = e.
If G (X, µ) is essentially free, and if there exists a fundamental domain, i.e. a Borel set W ⊂ X such that (g • W) g∈G is a partition of X up to measure zero, then G (X, µ) is called dissipative.On the other hand, if for every nonnegligible Borel set U ⊂ X there are infinitely many g ∈ G such that µ(gU ∩ U ) > 0, then we say the action G (X, µ) is conservative.
If G (X, µ) is an essentially free nonsingular action, there is a unique partition (up to measure zero) X = U ⊔ W of X into G-invariant Borel sets U and W such that the action G (U , µ) is conservative and G (W, µ) is dissipative.The actions G U and G W are called the conservative, resp.dissipative parts of the action G X. Finally note that an essentially free ergodic action must be conservative, except when the action is transitive, i.e. when µ is atomic and supported on a single G-orbit, which means that the action is isomorphic with the translation action G G.

Nonsingular Bernoulli actions
Suppose that X 0 is a standard measurable space and that G is a countable infinite group.For a family of equivalent probability measures (µ g ) g∈G on X 0 , consider the product probability space The action G (X, µ) given by (g By Kakutani's criterion for the equivalence of product measures [Kak48], the Bernoulli action G (X, µ) is nonsingular if and only if (1.2) holds, where H(µ, ν) denotes the Hellinger distance defined by where ζ is any probability measure on X with µ, ν ≺ ζ.
Recall that for any permutation ρ of G (finite or infinite) such that the induced transformation is nonsingular we have that ), with unconditional convergence a.e. on X.
We then have that

Maharam extension and associated flow
Let λ be the Lebesgue measure on R. The Maharam extension of a nonsingular automorphism ϕ ∈ Aut(X, µ) is the nonsingular automorphism ϕ ∈ Aut(X × R, µ × λ) that is given by Note that ϕ preserves the infinite measure dµ(x) × exp(−t)dλ(t).Also note that ϕ → ϕ is a continuous group homomorphism between the Polish groups Aut(X, µ) and Aut(X × R, µ × λ).
The translation action s • (x, t) = (x, t + s) commutes with every ϕ.For any nonsingular action G (X, µ) Krieger's associated flow (see [Kri76]) is defined as the action of R on the ergodic decomposition of the Maharam extension G X × R, which amounts to the action of R on Recall from the introduction how the type of a nonsingular group action G (X, µ) is defined and that, for essentially free, ergodic actions of amenable groups, the type and associated flow form a complete invariant of the action, both up to orbit equivalence and up to isomorphism of the crossed product factors L ∞ (X) ⋊ G.
3 Poisson flows and infinite divisibility: proof of Theorem C

Tail boundary flows
Recall from [CW88] the construction of the tail boundary flow as the Poisson boundary of a time-dependent Markov random walk on R with transition probabilities (µ n ) n∈N .Consider Choose a probability measure µ 0 on R that is equivalent with the Lebesgue measure and define the nonsingular maps

Define the von Neumann algebras
Then the tail boundary is defined as A = n≥0 A n .The translation action of R in the first variable defines an ergodic action of R on A, which is called the tail boundary flow.
We refer to [BV20, Section 2.3] for several basic results on tail boundary flows.
Tail boundary flows play a key role in this paper.When working with elements x of a product space as in (3.1), we always denote by x k or x n the natural coordinates of x.
Note that the tail boundary flow does not change if we permute the probability measures µ n .More precisely, if σ : N → N is any bijection, then the tail boundary flows of (µ n ) n∈N and (µ σ(n) ) n∈N are canonically isomorphic in the following way.Denoting by ( Ω, µ) the path space for the family (µ σ(n) ) n∈N , the map is a measure preserving bijection.We consider the von Neumann subalgebras We prove in this section two results on tail boundary flows that are of independent interest.First recall for future reference the following well known and easy result.For completeness, we include the short proof.The Hellinger distance H was defined in (2.1).We also make use of the total variation distance between probability measures µ, ν on a standard Borel space X: whenever ζ is a probability measure on X with µ, ν ≺ ζ.Note that for all probability measures µ and ν on X.
then the tail boundary flows of (µ n ) n∈N and (ν n ) n∈N are isomorphic.This conclusion holds in Define (Ω, µ) and (Ω n , µ n ) as in (3.1), so that the tail boundary A of (µ n ) n∈N is realized as the intersection of The Kakutani criterion for the equivalence of product measures implies that the identity map X n → X ′ n is a nonsingular isomorphism, inducing a * -isomorphism θ n : A n p n → C n p ′ n .By (3.3), the projections p n ∈ A and p ′ n ∈ D are increasing to 1.The * -isomorphisms θ n are compatible, so that there is a unique * -isomorphism θ : A → C satisfying θ(a)p ′ n = θ n (ap n ) for all a ∈ A and n ∈ N. By construction, θ conjugates the tail boundary flows of (µ n ) n∈N and (ν n ) n∈N .
We start by proving that such an identification of tail boundary flows also holds under a different approximation assumption, replacing the Hellinger distance by the Wasserstein 2-metric.We can do even slightly better by taking the Wasserstein 2-metric w.r.t. the metric on R given by d(x, y) = |T κ (x − y)|, where for κ > 0, we denote by T κ the cutoff function (3.4) Recall that a coupling between probability measures µ, ν on R is a probability measure η on R 2 such that, writing π 1 : R 2 → R : π 1 (x, y) = x and π 2 : R 2 → R : π 2 (x, y) = y, we have (π 1 ) * (η) = µ and (π 2 ) * (η) = ν.The set of all couplings between µ, ν is denoted as Γ(µ, ν).For every κ > 0, we then denote by then the tail boundary flows of (µ n ) n∈N and (ν n ) n∈N are isomorphic.
Note that the maps S m are nonsingular factor maps. Denote B = L ∞ (R × Ω, λ × η) and define We let R act by translation in the first variable and obtain in this way an ergodic action R A.
We identify the ergodic action R A with both the tail boundary flow of (µ n ) n∈N and the tail boundary flow of (ν n ) n∈N .
The first identification can be easily proved as follows and basically holds by definition.Writing and we get the measure preserving factor map By independence, we have that A ⊂ P * (D) and A m ∩ P * (D) = P * (C m ).Therefore, A = P * (C).
The second identification requires more work.We disintegrate the probability measures η n w.r.t. the second variable.We thus find probability measures η n,y on R such that we can view (Ω y , η y ) y∈Y as the disintegration of (Ω, η) w.r.t. the measure preserving factor map R : Ω → Y : R(x, y) = y.

Define the Borel functions
Also define the Borel set We already proved that F (y) < +∞ for ν-a.e.y ∈ Y .By van Kampen's version of Kolmogorov's three series theorem (see e.g.[Shi19, Theorem 3 in Section 4.2]), we have that ν(Y \ U ) = 0 and that for all y ∈ Y with F (y) < +∞, also and define the nonsingular factor map More precisely, we write y i , y m+1 , y m+2 , . ..) .
To prove the converse, fix F ∈ A. It follows that for ν-a.e.y ∈ Y , we may view the function F (•, y) as an element of the tail boundary for the measures (η n,y ) n∈N .By e.g.[BV20, Proposition 2.1], this tail boundary is given by the translation action R R and we find a unique This concludes the proof of the proposition.
In our applications of Proposition 3.2 in this paper, we will only need the following elementary estimate for the Wasserstein 2-distance.Assume that (β n ) n∈N are probability measures on R and assume that for every n ∈ N, we have which follows immediately by using the coupling η = ∞ n=1 p n (β n × δ tn ).Secondly, we prove a generalization of Orey's fundamental result in [Ore66, Theorem 3.1].He proved that if (µ n ) n∈N is a sequence of probability measures on R with uniformly bounded support, i.e. for which there exists a C > 0 such that µ n ([−C, C]) = 1 for all n ∈ N, the tail boundary flow is never properly ergodic: if ∞ n=1 Var µ n = +∞, the tail boundary flow is periodic, and if ∞ n=1 Var µ n < +∞, the tail boundary flow is given by the translation action R R.
The following result says that periodicity of the tail boundary flow already follows if we can find finite positive measures β n ≤ µ n such that the sequence (β n ) n∈N has uniformly bounded width and such that the sum of the properly normalized variances of β n is infinite.
More precisely, we prove the following and in particular, provide a functional analytic proof to [Ore66, Theorem 3.1].
Proposition 3.3.Let (µ n ) n∈R be a sequence of probability measures on R. Assume that β n are positive finite measures on R satisfying then the tail boundary flow of (µ n ) n∈R is periodic.
Note that, by convention, if certain β n are zero, we interpret the corresponding term in (3.7) as zero.The assumptions of Proposition 3.3 say in particular that, for each fixed n, the measure β n has a bounded support, so that its mean value and variance are well defined and finite.
Proof.Note that it suffices to prove the proposition assuming that all β n = 0. Indeed, it then follows that for I = {n ∈ N | β n = 0}, the tail boundary flow of (µ n ) n∈I is periodic, from which it follows that, a fortiori, the tail boundary flow of (µ n ) n∈N is periodic.
Assume that the sum in (3.7) is infinite.We prove that the tail boundary flow of (µ n ) n∈R is periodic.
Define the finite positive measures α n such that Denote by G the distribution function of a standard Gaussian random variable.By the Berry-Esseen theorem (see e.g.[Stro11, Theorem 2.2.17]), we get that for all k ∈ N and all t ∈ R. Take δ < 0 such that G(δ) = 1/3.Then, F k (δ) ≥ 1/12.Since σ k ≥ 40C and δ < 0, we conclude that Denote by µ 0 the probability measure on R given by dµ 0 (t) = (π(1 + t 2 )) −1 dt.Consider the Hilbert spaces and define the subspaces Note that T (z) = θ 1 (z) − θ 2 (z) for all z ∈ X 0 .We consider the associated measure preserving factor maps Define the isometries V : When ξ ∈ H N , we have W k (ξ) = V (ξ) for all k large enough.By density, we get that W k → V strongly.
Let F ∈ L ∞ (R × Ω) be a function that generates the tail boundary of (µ n ) n∈N .Since F belongs to the tail boundary algebra, we have and since each S k is a bounded function, we can define the bounded linear operators To prove this claim, note that the left hand side can be estimated, using the Cauchy-Schwartz inequality and (3.10), by So, the claim is proven.
We can then define the bounded convolution operator Note that by the Cauchy-Schwarz inequality and (3.10), We finally prove that for every ξ ∈ L ∞ (R × X) ⊂ K and every ξ ′ ∈ K, we have that Define the closed subspaces for all N ∈ N. By (3.11), the sequence R k j (ξ) 2 is bounded.To prove (3.12), we may thus assume that ξ ′ ∈ K N for some N .By (3.11), we also have that which tends to zero as N → ∞, uniformly in j.To prove (3.12), we may thus also assume that ξ ∈ K N .When ξ, ξ ′ ∈ K N and n k j > N , we have for all j, so that (3.12) follows.
We now return to our element F generating the tail boundary.By (3.9), we get that We thus conclude that F = η * F .Since η = δ 0 , it follows from the Choquet-Dény theorem (see [CD60]) that F is periodic in the first variable.So, the tail boundary flow is periodic.
In our applications of Proposition 3.3, we will use a few times the following elementary equality and estimate for every probability measure β on R.
In the following proposition, we develop this further and prove that, under the appropriate assumptions, the resulting sequence s n must itself be sparse: partitioning R into intervals I k of uniformly bounded length, we may assume that within each I k , the points s n lie close to a single element t k ∈ I k .
The proof of Proposition 3.4 is closely inspired by [GS83, Proposition 1.1] and [AW68, Lemma 8.6].Although this is not strictly needed for the rest of this paper, we use the opportunity to also deduce from Proposition 3.4 a more conceptual proof for the main result of [GS83] saying that every ITPFI factor of bounded type is isomorphic with an ITPFI 2 factor; see Theorem 3.5.Recall that, by definition, an ITPFI factor of bounded type is an infinite tensor product of matrix algebras M n k (C) with sup k n k < +∞.
Proposition 3.4.Let (µ n ) n∈N be probability measures on R. Let (J k ) k∈N be a family of disjoint subsets of N. Assume that for every k ∈ N, we are given p k , q k > 0 and an interval Assume that for each k ∈ N and n ∈ J k , we are given a probability measure If the tail boundary flow of (µ n ) n∈N is aperiodic, each J k is a finite set and there exist Note that there is a twofold difference between (3.14) and (3.15).In (3.15), the concentration points t k only depend on the interval I k and are thus the same for each n ∈ J k .On the other hand, the factor p k q k appearing in (3.15) is strictly smaller than the factor q k = (q k β n )(I k ) that we would get in (3.14).
Proof.For a fixed k ∈ N, we have that the measures (p k δ 0 + q k β n ) n∈J k have a uniformly bounded support in {0} ∪ I k .It thus follows from Proposition 3.3 that ) and β n is supported on I k , by (3.13), the left hand side is larger or equal than We can also apply Proposition 3.3 to all the finite measures q k β n ≤ µ n , with k ∈ N and n ∈ J k , because they have uniformly bounded width C. Defining for any k ∈ N and n ∈ J k , the point we get that For every k ∈ N, we consider the finitely many points (s n ) n∈J k in the interval I k .We denote by t k a "middle point".More precisely, if Since the tail boundary flow of (µ n ) n∈N is aperiodic, also the measures µ n * µ α k (n) with k ∈ N and n ∈ A k have an aperiodic tail boundary.By construction, Moreover, the measures on the left have their support in I k , with |I k | ≤ C for all k ∈ N. Again applying Proposition 3.3, we conclude that The mean value of ( and since p k ≤ 1, in combination with (3.16), we have proven that (3.15) holds.
Theorem 3.5.Every ITPFI factor of bounded type is isomorphic with an ITPFI 2 factor.
Proof.By induction, it suffices to prove that for every integer N ≥ 2, every ITPFI N +1 factor of type III 0 is isomorphic with the tensor product of an ITPFI N factor and an ITPFI 2 factor.
We denote in this proof by δ(a) the Dirac measure in a ∈ R. For every a ≥ 0, we define the probability measure For every a ∈ R N ≥0 , we define the probability measure and the state ψ a on M N +1 (C) by By diagonalizing states, any ITPFI N +1 factor can be written as the tensor product of a sequence (M N +1 (C), ψ an ) with a n ∈ R N ≥0 .By [CW88, Theorem 3.1], its flow of weights is precisely the tail boundary flow of the sequence (ρ(a n )) n∈N .
We thus fix such a sequence a n ∈ R N ≥0 , we assume that the tail boundary flow of is aperiodic and we prove that it is isomorphic with the tail boundary flow of a family of probability measures of the form ρ(b), b ∈ R N −1 ≥0 , and γ(c), c ≥ 0. For every k ∈ N and 1 ≤ i ≤ N , we denote We put p = (1 + N ) −1 and q k = p exp(−k).Fix i ∈ {1, . . ., N }.
For every n ∈ J 1,i , we have that p δ(0) + p exp(−a n,i ) δ(a n,i ) ≤ µ n and that the measures δ(a n,i ) are supported on the interval [0, 1].It follows from Proposition 3.3 and (3.13) that For every k ≥ 2 and n ∈ J k,i , we have that and that the measures at the left are supported on the interval [k − 1, k).For n ∈ J k,i , we have that exp(−a n,i ) ≤ exp(1) (N + 1) q k .By Proposition 3.4, we thus find b Defining b 1,i = 0 for all i ∈ {1, . . ., N } and summing over i ∈ {1, . . ., N }, it follows from (3.19) and (3.20) that Define the maps α n : {1, . . ., N } → N by α n (i) = k if and only if n ∈ J k,i .Define the probability measures Using the Wasserstein 2-distance, it follows from (3.21) and For every θ ∈ F N , we denote by θ the restriction of θ to {1, . . ., N − 1}.For every θ ∈ J , we apply Lemma 3.6 below to By Lemma 3.6, we thus find K θ , M θ ∈ N such that with the notation of (3.18), It then follows from (3.23) and Lemma 3.1 that the tail boundary flow of (µ n ) n∈N is isomorphic with the tail boundary flow of the family of measures This concludes the proof of the theorem.
Although the following lemma is an immediate consequence of [GS83, Lemma 2.2], it has never been stated in this very general form.
Lemma 3.6.Let P and Q be probability measures on R and α, β > 0. For every L ∈ N, we have that For all other integers k, m ≥ 0, we write µ(k, m) = 0.Then, We similarly write for all integers 0 ≤ k ≤ K and 0 ≤ m ≤ M , Parts (a) and (b) of the proof of [GS83, Lemma 2.2] are saying that H 2 (µ, µ ′ ) ≤ 2β.Then also

Poisson flows
Recall that to every finite positive measure µ on R is associated the compound Poisson distribution on R, which is defined as the probability measure By [GS83, Theorem 2.1], which we reproved as Theorem 3.5 above, every ITPFI factor of bounded type is isomorphic with an ITPFI 2 factor.So Poisson flows of positive type are also precisely the flows of weights of ITPFI factors of bounded type.Note that Theorem C is an immediate consequence of Theorems 3.8 and 3.9.
For Poisson flows with a nontrivial eigenvalue group, Theorem 3.9 was proven in [GH08, Proposition 7.1].It is easy to see that 2π/p is an eigenvalue of a Poisson flow R (Z, η) iff R (Z, η) is the tail boundary flow of a sequence of compound Poisson distributions with support in pZ.The main step in the proof of Theorem 3.9 is to show that in general, if R (Z, η) is a Poisson flow that is aperiodic, then we may realize R (Z, η) as the tail boundary flow of a sequence of compound Poisson distributions with very sparse support: at most one atom in each length one interval.
Before proving Theorem 3.8, we recall the following definition from [HOO74, Section 3], which we generalize in the natural way to multiple flows of locally compact abelian groups.We only use this concept for G = R and G = Z.Definition 3.10 ([HOO74, Section 3]).Let G be a locally compact second countable abelian group.For i ∈ {1, . . ., n}, let G (Z i , η i ) be a nonsingular action.Consider the direct sum G ⊕n and its natural action The action of G on A in the first variable (or, which is the same, in any of the other variables) is called the joint action of G (Z i , η i ).When G = R, we use the terminology joint flow, instead of joint action.
By construction, the tail boundary flow of the disjoint union (µ n ) n∈I⊔J of two countable infinite families of probability measures on R is the joint flow of the tail boundary flows of (µ n ) n∈I and (µ n ) n∈J .Also by construction, the flow of weights of a tensor product factor M 1 ⊗ M 2 is the joint flow of the flows of weights of M i .
As a final preparation to proving Theorem 3.8, we discuss the relation between tail boundary flows and induced actions.Assume that G is a locally compact second countable abelian group with closed subgroup H. Choose a probability measure ν on G/H that is equivalent with the Haar measure.Recall that any nonsingular action H (Y, η) has an induced action G (G/H × Y, ν × η), which is defined as follows.Choosing a Borel lift ψ : G/H → G, one defines the 1-cocycle Ω : G × G/H → H : Ω(g, x) = g + ψ(x) − ψ(g + x) and the induced action Another choice of lift ψ gives a cohomologous 1-cocycle and thus, an isomorphic induced action.
If (µ n ) n∈N is a family of probability measures on H, we have an associated tail boundary Haction.We can also view (µ n ) n∈N as a family of probability measures on G. Since the induction of the translation action H H is, by construction, the translation action G G, we have by definition that the tail boundary G-action of (µ n ) n∈N is isomorphic with the induction to G of the tail boundary H-action of (µ n ) n∈N .
Proof of Theorem 3.8.The cases of a trivial flow or a periodic flow are straightforward.By e.g.[BV20, Proposition 2.1], taking a > 0 and µ n = E(δ a ) for all n ∈ N, the tail boundary flow is the periodic flow R R/aZ.Similarly, taking a > 0 irrational and µ n = E(δ 1 + δ a ) for all n ∈ N, the tail boundary flow is trivial.So, it suffices to consider an almost periodic, aperiodic, ergodic flow.Such a flow is given by the translation action of R on a compact second countable abelian group L under a dense embedding π : R → L. We have to realize this flow as the tail boundary flow of a sequence of compound Poisson distributions E(µ n ) with µ n supported on R ≥0 .We start by reducing this problem to a similar question for Z-actions.
Fix a nontrivial character ω 0 ∈ L. Take t 0 ∈ R \ {0} such that ω 0 (π(t)) = exp(2πit/t 0 ) for all t ∈ R. So, ω 0 gives rise to a surjective, continuous group homomorphism θ : L → R/t 0 Z such that θ • π : R → R/t 0 Z is the natural quotient map.Define K = Ker θ as the kernel of θ.Note that the restriction of π to t 0 Z is a dense embedding of t 0 Z into K.Note that the translation action R L is the induction to R of the translation action t 0 Z K.By the discussion preceding this proof, it thus suffices to prove that this translation action t 0 Z K can be realized as the tail boundary action of t 0 Z associated with a sequence of compound Poisson distributions E(µ n ), where µ n are finite positive measures on t 0 N ⊂ t 0 Z.Since we can rescale everything with t 0 , we may assume that t 0 = 1.
Combining [CW83, Corollary 2.8] and [CW88, Theorem 3.4], we can take finitely supported probability measures (η n ) n∈N on Z such that the associated tail boundary flow is given by the translation Z K. Denote by η n the probability measure given by η n (U ) = η n (−U ).The tail boundary flow of ( η n ) n∈N is given by the action of Z on K by n • k = −n + k.Through the isomorphism k → −k, this flow is isomorphic with the original translation action Z K. Since the joint flow of Z K with itself is again Z K, the tail boundary flow of the measures (η n * η n ) n∈N is still given by Z K.
For every µ ∈ ℓ 1 (Z), we consider the Fourier transform We view K as a dense countable subgroup of T. The Fourier transform of η n * η n is a positive function.We replace η n by η n * η n and we may thus assume that the probability measures are finitely supported, with η n (ω) ≥ 0 for all n ∈ N and ω ∈ K, and with associated tail boundary flow Z K.
As each ω ∈ K is an eigenvalue of the translation action Z K, we know from [CW88, Theorem 4.2] that lim n→+∞ ∞ m=n η m (ω) = 1 for every ω ∈ K.

Choose an increasing sequence of finite subsets
Then define the probability measures α k on Z by Also, the tail boundary flow of (α k ) k∈N is still given by Z K.
For every k ∈ N, The tail boundary flow of (γ k ) k∈N is still given by Z K.
Define the compound Poisson distributions β k = E(kγ k ) and denote by Z X the tail boundary flow of (β k ) k∈N .We prove that Z X is isomorphic with Z K. Since Again by [CW88, Theorem 4.2], it follows that each ω ∈ K is an eigenvalue of the tail boundary flow of (β k ) k∈N .
Since exp(−k) is summable, by Lemma 3.1, the term exp(−k)δ 0 in the definition of β k = E(kγ k ) is negligible, so that Z X is isomorphic with the tail boundary flow of Denoting by Z Y the tail boundary flow of the sequence (ζ k ) k∈N , we conclude that Z X is the joint flow of Z K and Z Y .Realizing this joint flow inside L ∞ (K × Y ) with Z acting in the first variable, we conclude that the action Z X can be continuously extended to an action of K.This means that Z X is isomorphic with a factor of Z K. We have already seen that each ω ∈ K is an eigenvalue of Z X.So, Z X must be isomorphic with Z K.
We finally prove Theorem 3.9.Here and in the rest of this section, we often make use of Prokhorov's theorem (see [Pro53]) estimating the total variation distance between a binomial distribution and a Poisson distribution.Given n ∈ N, p ∈ [0, 1] and λ ≥ 0, define the binomial distribution β and Poisson distribution π by In the version of [BH83, Theorem 1], Prokhorov's theorem says that Proof of Theorem 3.9.The trivial flow and every periodic flow arise as the flow of weights of an ITPFI 2 factor and also arise as a Poisson flow of positive type.We thus only consider the aperiodic case.
First assume that R X is the flow of weights of an ITPFI 2 factor and that R X is aperiodic.As we have seen in the proof of Theorem 3.5, we find for all k ∈ N, elements b X is isomorphic with the tail boundary flow of the sequence γ(b k ) * L k , where γ(b) is defined by (3.18).By (3.28), we get that ) is summable, it follows from Lemma 3.1 that R X is a Poisson flow of positive type.
Conversely, assume that R X is the Poisson flow of positive type defined by the sequence of probability measures µ n = E(η n ), where each η n is a finite positive measure supported on (0, +∞).We may assume that R X is aperiodic.Taking integers L n ≥ η n (R), we can replace µ n by E(η n /L n ), each repeated L n times.We may thus assume that λ n := η n (R) ≤ 1.
For every fixed C > 0, we have that .Since E(η n ) is the convolution of the measures E(ρ n ) and E(η n,k ), n, k ∈ N, we conclude that R X also is the tail boundary flow of the union of the probability measures E(ρ n ), n ∈ N, and E(η n,k ), n, k ∈ N. By (3.30) and (3.24), we have ∞ n=1 Var(E(ρ n )) < +∞.By e.g.[BV20, Proposition 2.1], the tail boundary flow of (E(ρ n )) n∈N is given by the translation action R R, so that R X is the tail boundary flow of the measures E(η n,k ), n, k ∈ N.
For every fixed k ∈ N, it follows from (3.30) that X also is the tail boundary flow of the measures E(ζ k /L k ), each repeated L k times, and since exp(−1) Denote by W 2 the Wasserstein 2-distance.By (3.6), we have that For every j ≥ 1, we have that We conclude that By Proposition 3.2, R X is also the tail boundary flow of the sequence By (3.28), we have By (3.28), we have Z is isomorphic with the tail boundary flow of the probability measures Conversely, assume that (µ n ) n∈N is a sequence of probability measures whose support consists of two points and that satisfy Var µ n ≤ C for all n ∈ N. Denote by R Z their tail boundary flow.We have to prove that R Z is a Poisson flow.We may thus assume that R Z is aperiodic.Since translating the measures µ n does not change the tail boundary flow, we may assume that µ n (0) ≥ 1/2 for all n ∈ N and we denote by d n ∈ R the other atom of µ n .Write Arguing as in the proof of Theorem 3.9, it follows from Proposition 3.3 that and it follows from Proposition 3.4 that we find for every By (3.31), the tail boundary flow of the measures (µ n ) n∈J −1 ∪J 0 ∪J 1 is given by the translation action R R, so that these measures may be ignored.Defining for every k ∈ I and n ∈ J k the probability measure η It thus follows from Proposition 3.2 that R Z is the tail boundary flow of the measures η n , k ∈ I, n ∈ J k .Write λ k = n∈J k p n .By [BH83, Theorem 1], we have for every k ∈ I that where we used that d n ∈ [k, k + 1) if n ∈ J k .Since the right hand side is summable, it follows from Lemma 3.1 that R Z is also the tail boundary flow of the Poisson distributions (E(λ k δ b k )) k∈I and thus is a Poisson flow.
The positive case is proven entirely analogously.

Infinitely divisible flows
Definition 3.12.We say that a flow R (Z, η) is infinitely divisible if for every integer L ≥ 1, there exists a flow R (Z 1 , η 1 ) such that R (Z, η) is isomorphic with the joint flow of L copies of R (Z 1 , η 1 ).
Every tail boundary flow of a sequence of infinitely divisible distributions is a Poisson flow.
Proof.If R Z is the tail boundary flow of the sequence (E(µ n )) n∈N , where µ n is a sequence of finite positive measures on R, and if L ≥ 1 is an integer, we can define R Z 1 as the tail boundary flow of the sequence (E( Since compound Poisson distributions E(µ) are weakly dense in the set of infinitely divisible distributions (see e.g.[Stro11, Theorem 3.2.7]), the second statement follows directly from Lemma 3.14 below.
Lemma 3.14.Let F ⊂ Prob(R) with weak closure F .Every ergodic flow that can be obtained as the tail boundary flow of a sequence in F can also be obtained as the tail boundary flow of a sequence in F.
Proof.Let (µ n ) n∈N be a sequence in F .Denote by ν n the uniform probability measure on the interval [−1/n, 1/n], so that Var ν n = n −2 /2 is summable.By e.g.[BV20, Proposition 2.1], the tail boundary flow of the sequence (ν n ) n∈N is the translation action R R. Therefore, (µ n ) n∈N and (µ n * ν n ) n∈N give rise to isomorphic tail boundary flows.
Since F is weakly dense in F and since ν n is absolutely continuous, we can choose µ Proof.The convolution products of measures of the form σ λ,a are precisely the compound Poisson distributions E(µ) where µ is a finite positive measure with finite support.These E(µ) are weakly dense in the set of all compound Poisson distributions.The conclusion thus immediately follows from Lemma 3.14.
Remark 3.16.Because of Proposition 3.13, it is tempting to speculate that every infinitely divisible flow is a Poisson flow, at least if the flow is assumed to be approximately transitive (and thus, the tail boundary flow of some sequence of probability measures, by [CW88, Theorem 3.2]).We have however no idea how to prove such a statement.
Going back to Theorem 3.9, it is also unclear whether every Poisson flow is automatically of positive type.Because of Theorem 3.9, this question is equivalent with the following seemingly innocent, but highly tantalizing problem: if R α Z is the flow of weights of an ITPFI 2 factor, does it follow that the reverse flow β t (z) = α −t (z) also is the flow of weights of an ITPFI 2 factor?
Combining both open problems, it is tempting to speculate that the ITPFI 2 factors are precisely the injective factors M that are infinitely divisible, in the sense that for every integer L ≥ 1, there exists a factor N such that M ∼ = N ⊗L , or to speculate that at least, infinite divisibility characterizes the ITPFI 2 factors among the ITPFI factors.

Nonsingular Bernoulli shifts: proof of Theorem A
The goal of this section is to prove the following more precise formulation of Theorem A. Since we want to describe absolutely general Bernoulli shifts, the formulation becomes a bit lengthy, because we have to deal with the less interesting cases that may arise where the space has atoms or the action is dissipative.
We denote by S the group of finite permutations of the countable set Z and let S act on Theorem 4.1.Let X 0 be any standard Borel space and let (µ n ) n∈Z be any family of equivalent probability measures on X 0 such that the Bernoulli shift Z (X, µ) = n∈Z (X 0 , µ n ) is nonsingular.
Then precisely one of the following statements holds.
1.There exists an atom b ∈ X 0 with n∈Z (1 − µ n ({b})) < +∞.Define a ∈ X by a n = b for all n ∈ Z.Then, a is an atom in X that is fixed by Z.The action of Z on X \ {a} is essentially free and dissipative.
2. The action Z (X, µ) is essentially free and dissipative.
3. The space (X, µ) is nonatomic and there exists a Borel set C 0 ⊂ X 0 of positive measure, unique up to a null set, such that n∈Z (1 − µ n (C 0 )) < +∞ and such that the following holds.
• The action Z C Z 0 is a weakly mixing Bernoulli shift and its associated flow is infinitely divisible.Moreover, the permutation action S C Z 0 is ergodic and for every ergodic pmp action Z (Y, ν), the actions Z C Z 0 , Z C Z 0 × Y and S C Z 0 have the same associated flow.
• The action Z X \ C Z 0 is essentially free and dissipative.
As already suggested by the formulation of Theorem 4.1, we again exploit the relation between a Bernoulli shift and the action S (X, µ).This method was discovered in [Kos18, Dan18] and has been further developed in [BKV19,BV20].
Because of Theorem 4.1, to study nonsingular Bernoulli shifts in their full generality, it suffices to consider Bernoulli shifts that are conservative.Also, given any conservative Bernoulli action We prove this claim using essentially the same argument as the one given in [BKV19, Theorem 3.3].Indeed if such sequences do not exist, there are δ > 0 and N ∈ N such that for all n ≤ −N, m ≥ N we have that H(µ n , µ m ) > δ.Define the 1-cocycle for every k ≥ 2N .Therefore k∈Z exp − c k 2 /2 < +∞ and it follows from [VW17, Theorem 4.1] that the action Z (X, µ) is dissipative, which is in contradiction with our assumptions.
Let n k → −∞ and m k → +∞ be sequences such that (4.1) holds.We may that n k < 0 < m k for all k ∈ N. Let α ∈ Aut(X, µ) denote the shift by one, i.e. (α(x)) k = x k−1 .To prove the lemma, it suffices to show that α belongs to the closure of S.
For each k ∈ N we define the permutation σ k ∈ S by We will show that θ k (F ) − F 2 → 0 for all F in a total subset of L 2 (X, µ), so that β k → id as k → +∞.This then concludes the proof of the lemma.
Take F ∈ L ∞ (X, µ) depending only on the coordinates x n , for |n| ≤ N , for some N ∈ N.With unconditional convergence almost everywhere we have that ) is summable and by our choice of n k , m k .We see that for n k < −N and m k > N we have that which converges to 0 as k tends to infinity.
Lemma 4.5.Let Z (X, µ) = n∈Z (X 0 , µ n ) be a nonsingular Bernoulli shift that is essentially free and not dissipative.Let C ⊂ X denote its conservative part.Let Z (Y, ν) be any ergodic pmp action and consider the diagonal product action Z X × Y .Then the following holds.

The Maharam extensions satisfy
Proof.Let α k ∈ Aut(X, µ) denote the translation by k, i.e. (k • x) m = x m−k .We proceed as in the proof of [BKV19, Lemma 3.1] and show that the dissipative part D ⊂ X given by It suffices to show that D is invariant under the permutation σ n ∈ S that interchanges the coordinate 0 and n.Fix n ∈ Z \ {0}.For each η > 0 and each k ∈ Z we define For every η > 0 we have that This means that the set has positive measure, µ(A η ) > 0. From the nonsingularity of Z (X, µ) it follows that µ(α m (A η )) > 0 for every m ∈ Z. Since x ∈ α m (A η ) if and only if x k ∈ X 0 \ A η k−m for all k ∈ Z, we conclude that k∈Z µ k+m (A η k ) < +∞ for every m ∈ Z. Similarly we have that k∈Z µ k+m (B η k ) < +∞ for every η > 0 and every m ∈ Z. Write For m ∈ Z, denote π m : X → X 0 for the coordinate projection π m (x) = x m .For any m ∈ Z we have that Therefore we have that Note that By the definition of W 1 x i and the sets A η k and B η k in (4.2), we have that 1/16 ≤ D k,k+n (x 0 , x n ) ≤ 16 whenever k / ∈ W 1 x 0 ∪ W 1 xn .So it follows from (4.4) and (4.5) that D is invariant under σ n .We now prove point 2. Let G (Y, ν) be an ergodic pmp action.One can repeat the proof of [BKV19, Lemma 3.1], making use of the ergodic theorem established in [Dan18, Theorem A.1], to conclude that In combination with (4.6), we get that proving the second statement of the Lemma.
Remark 4.6.Except for the point where we invoke Lemma 4.4, the proof of Lemma 4.5 remains valid for a nonsingular Bernoulli action G g∈G (X 0 , µ g ) of any countable infinite amenable group G, as long as also the right Bernoulli action is nonsingular, e.g. when G is abelian.However, we were unable to prove an analogue of Lemma 4.4 for arbitrary abelian groups.That is the main reason why this section is restricted to the group of integers.Note that it is nevertheless straightforward to generalize our results from Z to virtually cyclic abelian groups.
Proof of Theorem 4.1.First assume that (X, µ) admits an atom d ∈ X.Then d n ∈ X 0 is an atom for every n ∈ Z and n∈Z (1 − µ n ({d n })) < +∞.Writing U = {x ∈ X | x n = d n for all but finitely many n ∈ Z } , it follows that µ(X \ U ) = 0. Since the shift is nonsingular, the set 1 • U ∩ U has measure 1.We thus find N ∈ N such that d n−1 = d n for all |n| ≥ N .
There are now two possibilities.Either we find an atom b ∈ X 0 such that d n = b for all |n| ≥ N , or we find two distinct atoms b, c ∈ X 0 such that d n = b for all n ≥ N and d n = c for all n ≤ −N .
In the first case, we get that n∈Z (1 − µ n ({b})) < +∞ and we define the atom a ∈ X by a n = b for all n ∈ Z.Clearly, g • a = a for all g ∈ Z.We define for every k ∈ N, the Borel set W k = {x ∈ X | x n = b whenever |n| ≥ k }.We have k∈N W k = U so that this set has a complement of measure zero.Also, g • (W k \ {a}) ∩ W k = ∅ whenever g ∈ Z and |g| > 2k.So, for every k ∈ N, the set W k \ {a} belongs to the dissipative part of the essentially free action Z X. Taking the union over k, we conclude that Z X \ {a} is essentially free and dissipative.
In the second case, we define for every k ∈ N, the Borel set X is essentially free and dissipative.
For the rest of the proof, we may thus assume that (X, µ) is nonatomic.Then the Bernoulli shift Z (X, µ) is essentially free, by [BKV19, Lemma 2.2].If Z (X, µ) is dissipative, the conclusion of point 2 holds.It now remains to consider the case where the conservative part C ⊂ X of Z (X, µ) has positive measure.We have to prove the structural result in point 3 of the theorem.
Note that C is Z-invariant.By Lemma 4.5, C is also S-invariant.We claim that for any integer p ≥ 2 we have that To prove this claim, fix p ≥ 2 and define Y = Z/pZ, equipped with the normalized counting measure ν.We let Z act on Y by translation.From Lemma 4.5 we know that , and this is exactly the statement (4.7).
For any integer p ≥ 2 and i ∈ {0, 1 . . ., p − 1}, we write (Z p,i , ν p,i ) = n∈i+pZ (X 0 , µ n ).We identify and we obtain measure preserving factor maps π p,i : X → Z p,i .For each i ∈ {0, 1, . . ., p − 1}, we have a Bernoulli action pZ Z p,i and the factor maps π p,i are pZ-equivariant.Let S p,i denote the group of finite permutations of i + pZ.We also have a nonsingular action S p,i Z p,i and π p,i is S p,i -equivariant as well.
For i ∈ {0, 1, . . ., p − 1}, write α i ∈ Aut(X, µ) for the shift by i.There is a natural nonsingular isomorphism θ p,i : Z p,0 → Z p,i such that θ p,i • π p,0 = π p,i • α i .We start by using (4.7) for p = 2 to show that L ∞ (C) Z is discrete as a von Neumann algebra.To simplify the notation, we will drop the index p for this special case p = 2.
Let E 0 ⊂ Z 0 be a Borel set such that (π 0 ) * µ| C ∼ ν 0 | E 0 .Then E 0 is uniquely determined up to a null set.As C is Z-invariant and π 0 is a 2Z-equivariant factor map, E 0 is 2Z-invariant.Similarly E 0 is S 0 -invariant.Since 2Z ⊂ Z has finite index, the action 2Z C is conservative.Therefore also 2Z E 0 is conservative and it follows that E 0 is contained in the conservative part of the Bernoulli action 2Z Z 0 .By Lemma 4.5 we have that By the equivariance of θ 1 we also have that E 1 is 2Z-and S 1 -invariant, and that 2Z and we apply (4.7) to conclude that F • π 0 is Z-invariant.Using that θ 1 • π 0 = π 1 • α 1 and viewing X = Z 0 × Z 1 , we can also express this as The equality (4.9) holds for any F ∈ L ∞ (E 0 ) 2Z , forcing L ∞ (E 0 ) 2Z to be discrete.Similarly, we see that L ∞ (E 1 ) 2Z is discrete as well.
Using once more the identification X = Z 0 × Z 1 , we have that C ⊂ E 0 × E 1 .Therefore, by Lemma 4.5, we have that Take a Z-invariant Borel set U ⊂ X with µ(U ) > 0 such that 1 U is a minimal projection in L ∞ (C) Z .So, Z U is ergodic.We prove that U is of the form U = C Z 0 for some C 0 ⊂ X 0 .For any integer p ≥ 2 and i ∈ {0, 1, . . ., p−1} define the Borel set U p,i by (π p,i ) * µ| U ∼ (ν p,i )| U p,i .First of all, note that U p,i is pZ-invariant and S p,i -invariant.By (4.7) the action pZ U is ergodic, so that also pZ U p,i is ergodic.Since we can view pZ Z p,i as a Bernoulli action, it follows from Lemma 4.5 that S p,i U p,i is ergodic as well.
Using the identification (4.8), we have that U ⊂ U p,0 × U p,1 × • • • × U p,p−1 for any p ≥ 2. As U is invariant under the subgroup S p,0 × S p,1 × . . .S p,p−1 and as S p,i U p,i acts ergodically for each i ∈ {0, 1, . . ., p − 1}, we have that U = U p,0 × U p,1 × . . .U p,p−1 mod µ, for every p ≥ 2. (4.10) Let n ∈ N and let A n ⊂ L ∞ (X) denote the subalgebra of elements only depending on the variables x j , for −n ≤ j ≤ n, and let E n : L ∞ (X) → A n be the unique conditional expectation preserving the measure µ.For any n ∈ N we apply the decomposition (4.10) to p = 2n + 1.Since the numbers {j | −n ≤ j ≤ n} are distinct representatives of the elements of Z/(2n + 1)Z, there exist a n,j ∈ L ∞ (X 0 , µ j ) such that 0 ≤ a n,j ≤ 1 a.e. for every −n ≤ j ≤ n.
Expressing that For each j ∈ Z, letting m → +∞, we have that a m,j is a sequence in L ∞ (X 0 , µ j ) such that 0 ≤ a m,j ≤ 1 for all m.Similarly, for a fixed n ∈ N, we have that n<|j|≤m µ j (a m,j ) ∈ [0, 1] for every m ≥ n.
Thus we can choose a subsequence m k → +∞ such that a m k ,j → b j weakly for every j ∈ Z and b j ∈ L ∞ (X 0 , µ j ) satisfying 0 ≤ b j ≤ 1, and such that n<|j|≤m k µ j (a m k ,j ) → λ n for every n ∈ N for some λ n ∈ [0, 1].The equality (4.11) implies that As E n (1 U ) is nonzero for every n, we see that λ n and b j are nonzero for every n ∈ N and j ∈ Z.
Expressing once more that E n • E m = E n for m ≥ n, we obtain that which shows that the infinite product |j|>n µ j (b j ) converges to a nonzero limit for each n ∈ N.
Let λ ∈ [0, 1] be any limit point of the sequence λ n .Using that E n (1 U ) → 1 U strongly as n → +∞, we see that the infinite product of the b j converges and that we have an equality Together with the fact that U is Z-invariant, this implies that there is a Borel set By construction, the action Z X \ C Z 0 is dissipative.It follows that C = C Z 0 = U .We have chosen U such that Z U is ergodic, thus it follows from Lemma 4.5 that S C Z 0 is ergodic, that Z C Z 0 is weakly mixing, and that for any ergodic pmp action Z (Y, ν) the nonsingular actions Z C Z 0 , S C Z 0 and Z C Z 0 × Y have the same associated flow.It remains to prove that this flow is infinitely divisible.For this remaining part of the proof, we may replace X 0 by C 0 and thus assume that C 0 = X 0 .
To prove this, let p ≥ 1 be an integer.We use the notation introduced in (4.8).Let S p,i ⊂ S denote the subgroup of finite permutations of i + pZ.We have that (4.12) For each i ∈ {0, 1, . . ., p − 1}, we write Γ p,i for the group Γ p,i = pZ, acting naturally on Z p,i .We can view the action Γ p,i Z p,i as a nonsingular Bernoulli action, which is a factor of the conservative nonsingular Bernoulli action pZ (X, µ).Therefore Γ p,i Z p,i is conservative and by Lemma 4.4, we have that L Each Γ p,i is a copy of pZ and the diagonal copy of pZ inside Γ p,0 × • • • × Γ p,p−1 acts on X by the Bernoulli action pZ X. Continuing the chain of inclusions (4.13), we obtain where the last equality follows from point 2 of Lemma 4.5, applied to the ergodic pmp action Z Y = Z/pZ.Combining (4.12), (4.13) and (4.14), we see that all inclusions must in fact be equalities.Put (D p , η p ) = n∈pZ (X 0 , µ n ).For each i ∈ {0, 1, . . ., p−1}, the action Γ p,i Z p,i is conjugate with pZ D p .From the equality it then follows that the associated flow of Z X is the joint flow of p copies of the associated flow of pZ D p .This concludes the proof of the theorem.
We end this section by proving Theorem 4.3.We first need the following lemma.
Lemma 4.7.Let X 0 be a standard Borel space equipped with a sequence of equivalent probability measures µ n .Let S denote the group of finite permutations of N and let S 1 ⊂ S be the subgroup fixing 1 ∈ N. Consider the nonsingular group actions S Then S (X, µ) is ergodic and the following holds.
1. S (X, µ) is of type II 1 if and only if there exists a probability measure ν ∼ µ 1 on X 0 such that ν N ∼ µ.

S
(X, µ) is of type II ∞ if and only if there exists a σ-finite measure ν ∼ µ 1 on X 0 and Borel sets U n ⊂ X 0 such that ν(U n ) < +∞ for all n ∈ N and such that Note that Lemma 4.7 strongly resembles [BV20, Theorem 3.3].There is however an important difference: in [BV20, Theorem 3.3], it is part of the hypotheses that the Radon-Nikodym derivatives dµ n /dµ 0 satisfy a certain boundedness condition.We do not make such an assumption, because we will use Lemma 4.7 in the context of totally arbitrary Bernoulli shifts.As a compensation, we make an ergodicity assumption on the permutation action.Thanks to Theorem 4.1, this ergodicity assumption will hold automatically when the Bernoulli shift Z X is conservative.
When X 0 is a finite set and (µ n ) n≥1 are equivalent probability measures on X 0 , there is a necessary and sufficient ergodicity criterion for the nonsingular permutation action S As S 1 acts ergodically on (Z, η), we see that F essentially only depends on the coordinate x 1 .But as F is S-invariant, it follows that F essentially only depends on the coordinate x 2 , thus F must be essentially constant.So the action S (X, µ) is ergodic.
If x, y ∈ X are elements that differ in only finitely many coordinates, we write Assume that the action S (X, µ) is semifinite.Then there exists a Borel map F : X → R such that α(x, σ(x)) = F (x) − F (σ(x)) for every σ ∈ S and a.e.x ∈ X.
(4.15) Define ( X, µ) = (X 0 × X 0 × Z, µ 1 × µ 1 × η), by doubling the first coordinate and consider the map For each σ ∈ S 1 , we have that H(x, x ′ , z) = H(x, x ′ , σ(z)) for a.e.(x, x ′ , z) ∈ X.As the action S 1 (Z, η) is ergodic, H is essentially independent of the z-variable.Therefore there exists a Borel map L : X 0 × X 0 → R such that H(x, x ′ , z) = L(x, x ′ ) for a.e.(x, x ′ , z) ∈ X.Let z ∈ Z be an element that witnesses this equality a.e. and put β(x) = F (x, z).So we have found a Borel map β : X 0 → R such that F (x) − F (y) = β(x 1 ) − β(y 1 ) , when x and y are unequal only in the first coordinate.For n ≥ 2, using (4.15) and the element σ n ∈ S flipping the elements 1 and n, we see that Let R Ω be the equivalence relation on X × R that is given by (x, t) ∼ (y, s) if and only if x and y differ only in finitely many coordinates and s−t = Ω(x, y).Then the flow R L ∞ (X ×R) R Ω is isomorphic with the tail boundary flow associated to the sequence of probability measures (α n + β) * µ n .By (4.16) we have that Ω(x, y) = F (x) − F (y) for x, y ∈ X that differ only in finitely many coordinates.We conclude that the tail boundary flow associated to (α n + β) * µ n is isomorphic with the translation action R R.
We again use the cutoff function T κ : R → R for κ > 0, as defined in (3.4).By [BV20, Proposition 2.1] there exists a sequence t n ∈ R such that ∞ n=1 X 0 T κ (α n (x) + β(x) − t n ) 2 dµ n (x) < +∞, (4.17) for every κ > 0. Define the σ-finite measure ν ∼ µ 1 by dν/dµ 1 = exp(−β).If ν is finite, then we can add a constant to β, so that ν becomes a probability measure.Then (4.17) still holds with a potentially different sequence t n ∈ R. Thus we may assume that ν is either infinite, or a probability measure.Define the sets By (4.17 σ-finite measure ν and Borel sets U n ⊂ X 0 satisfying the conditions of the second point of the theorem. Assume we are given such ν and U n .Then, as in (4.20), the sum is unconditionally convergent a.e.By (4.21) the map (x, t) → t − G(x) is invariant under the Maharam extension of Z. Also, it is R-equivariant.
5 Nonsingular Poisson suspensions: proof of Proposition D We start this section by recalling the construction of the Poisson suspension.For a detailed treatment, we refer to [Roy08,DKR20].Let (X 0 , µ 0 ) be a σ-finite standard measure space.We write B 0 = {A ⊂ X 0 | A is Borel and µ 0 (A) < +∞}.To (X 0 , µ 0 ), one associates a standard probability space (X, µ) and random variables P A : X → {0, 1, 2, . . .} for every A ∈ B 0 such that the following holds.
1.The random variable P A is Poisson distributed with intensity µ 0 (A).
2. If A, B ∈ B 0 are disjoint, then P A and P B are independent random variables and we have that P A∪B = P A + P B .
3. The family (P A ) A∈B 0 separates the points of X.
These three properties uniquely characterize (X, µ) and the random variables (P A ) A∈B 0 .The probability space (X, µ) is called the Poisson suspension over the base space (X 0 , µ 0 ).
By the functoriality of this construction, every measure preserving Borel automorphism θ : X 0 → X 0 gives rise to an essentially unique, measure preserving Borel automorphism θ : X → X such that for every A ∈ B 0 , we have P A ( θ(x)) = P θ −1 (A) (x) for µ-a.e.x ∈ X. (5.1) In [DKR20, Theorem 3.3], it was discovered that a nonsingular Borel automorphism θ : X 0 → X 0 gives rise to a nonsingular Borel automorphism θ : X → X satisfying (5.1) if and only if For completeness, we include below a short proof of one implication, namely that every θ satisfying (5.2) has a suspension θ.This proof is essentially taken from [DKR20], but presented in a more direct way.
So, whenever G (X 0 , µ 0 ) is a nonsingular action of a locally compact second countable (lcsc) group such that sup g∈K d(gµ 0 )/dµ 0 − 1 2 < +∞ for every compact K ⊂ G, (5.3) we have an essentially unique nonsingular action G (X, µ) characterized by P A (g • x) = P g −1 •A (x), which is called the Poisson suspension action.
The main goal of this section is to prove Proposition D. We actually prove the following stable version, that also considers the associated flow of the diagonal action G X × Y for any ergodic pmp action G (Y, ν).
Proposition 5.1.Let G be any lcsc group that does not have property (T ).Let R (Z, ζ) be any Poisson flow.Then G admits a nonsingular action G (X 0 , µ 0 ) of which the Poisson suspension G (X, µ) is well-defined, weakly mixing, essentially free and such that for any ergodic pmp action G (Y, ν) the diagonal action G X × Y has associated flow R Z.
Proposition 5.1 is sharp in the following sense: by [DKR20, Theorem G], if G has property (T), then every Poisson suspension action admits an equivalent invariant probability measure.This follows by applying the Delorme-Guichardet theorem (see e.g.[BHV08, Theorem 2.12.4]) to the 1-cocycle g → d(gµ 0 )/dµ 0 − 1 with values in the Koopman representation of G (X 0 , µ 0 ).
Before proving Proposition 5.1, we introduce some further background, based on [DKR20].In particular, we give a short proof that every nonsingular automorphism θ : X 0 → X 0 satisfying (5.2) admits a Poisson suspension θ.This proof is essentially taken from [DKR20], but since our approach is direct and short, we include it here for convenience of the reader.
Write H = L 2 R (X 0 , µ 0 ) and denote by H n ⊂ H ⊗n the closed subspace of symmetric vectors (i.e.invariant under the action of the symmetric group S n on H ⊗n ).The key point is that there is a canonical isometric isomorphism between the symmetric Fock space F s (H) of H and L 2 R (X, µ).This isomorphism U is defined as follows.For every ξ ∈ H, denote by exp(ξ) ∈ F s (H) the usual exponential given by exp(ξ) = 1 ⊕ ∞ n=1 1 n! ξ ⊗n .

. 24 )
Definition 3.7.We call a Poisson flow any ergodic flow R (Z, η) that arises as the tail boundary flow of a sequence of compound Poisson distributions on R. If these compound Poisson distributions can be chosen with support in R ≥0 , we call R (Z, η) a Poisson flow of positive type.Note that compound Poisson distributions and tail boundary flows make sense on any locally compact second countable abelian group G, leading to the concept of a Poisson G-action, which we will use for G = R and G = Z.In Remark 3.16, we discuss the relation between general Poisson flows and Poisson flows of positive type.The two main result of this section are the following.Theorem 3.8.Every ergodic almost periodic flow is a Poisson flow of positive type.Theorem 3.9.The Poisson flows of positive type are precisely the flows of weights of ITPFI 2 factors.
dense, there are arbitrarily large positive integers n ∈ U k .Since the measures α k have finite support, we can choose m k ∈ N large enough such that m k ∈ U k and such that the translated measures defined by γ k (V) = α k (V − m k ) have their support in N. Since m k ∈ U k , using (3.26), we get that |1 − γ k (ω)| ≤ 2k −3 for every ω ∈ F k .(3.27) .29) Since exp(−λ n ) ≥ exp(−1) and η n ((0, C]) ≤ 1 for all n ∈ N, it follows from (3.29) and (3.13) that for every C > 0, ∞ n=1 (0,C] x 2 dη n (x) < +∞ .(3.30) Denote ρ n = η n | (0,1] and, for all k ∈ N, η n,k = η n | (k,k+1] the sequence exp(−b k ) is summable.It then follows from Lemma 3.1 that R X is the tail boundary flow of the measures (γ(b k ) * M k ) k∈N and hence, is isomorphic with the flow of weights of an ITPFI 2 factor.The proof of Theorem 3.9 relied on Prokhorov's (3.28).Using the full force of [BH83, Theorem 1], we can also prove the following result.Proposition 3.11.The Poisson flows are exactly the tail boundary flows of sequences (µ n ) n∈N where the support of all µ n consists of two points and sup n∈N Var µ n < +∞.The Poisson flows of positive type are exactly the tail boundary flows of such sequences (µ n ) n∈N with support a n < b n and µ n (a n ) ≥ µ n (b n ), and sup n∈N Var µ n < +∞.Some bound on the variances Var µ n must be imposed in order to get a Poisson flow.Combining [GSW84, Theorem 2.1] and Proposition 3.13 below, it follows that the tail boundary flow of the sequence of probability measures (δ 0 + δ 8 n )/2 n∈N is not a Poisson flow.Proof.First assume that R Z is a Poisson flow.In most of the proof of Theorem 3.9, we did not use that the measures are supported on the positive real line.Writing I = Z \ {−1, 0}, we find for every k ∈ I, elements b k ∈ (k, k + 1] and constants λ k > 0 such that R Z is isomorphic with the tail boundary flow of the family (E(λ k δ b k )) k∈I .Choose for every k ∈ I an integer M k ≥ 1 such that M −1 k λ k ≤ 2 −|k| .Define for k ∈ I, the probability measure give rise to isomorphic tail boundary flows, with the latter being isomorphic to the tail boundary flow of (µ ′ n ) n∈N .Define for every λ > 0 and a ∈ R the standard Poisson distribution with support {ka | k = 0, 1, 2, . ..} given by σ λ,a ({ka}) = exp(−λ) λ k k! for all k ∈ {0, 1, 2, . ..}. (3.33) Proposition 3.15.Every Poisson flow is the tail boundary flow of a sequence (σ λn,an ) n∈N with λ n > 0 and a n ∈ R.

(4. 6 )
Write λ for the Lebesgue measure on R. As the Maharam extension map Aut(X, µ) → Aut(X × R, µ × λ) : ϕ → ϕ is continuous, it follows from the S-invariance of C and Lemma 4.4 that
terms of the measures µ n , see [AP77, Theorem 1.6].However, when X 0 is infinite, only sufficient conditions are known, see [AP77, Theorems 1.8 & 1.12].The measure ν appearing in statement 2 of Lemma 4.7 is either finite, or infinite.Of course, if ν is finite, the condition ν(U n ) < +∞ is automatically fulfilled.Similarly, when ν is infinite, the conditions ν