Sorin Popa, Roman Sasyk
We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to Π} (X_0, μ_0)_g$ for $(X_{0},μ_{0})$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.
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We conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III$_1$ if and only if the first $L^2$-cohomology of $G$ is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III$_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity.
This paper has been withdrawn by the author.
Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if $G_1,G_2$ are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over $G_1$ is stably orbit equivalent to every nontrivial Bernoulli shift over $G_2$. This answers a question of S. Popa.
For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced crossed products $A^{\otimes G}\rtimes_rG$ of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for finite-dimensional C*-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the $K$-theory of reduced C*-algebras of wreath products $H\wr G$ for large classes of groups $H$ and $G$. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li…
We introduce a non-commutative extension of Tsirelson-Vershik's noises, called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory. Such shifts are, in particular, capable of producing Arveson's product system of type I and type II. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (`classical') stationary flows for Tsirelson-Vershik's noises is now played b…
We show that for a countable exact group, having positive first $\ell^2$-Betti number implies proper proximality in this sense of \cite{BoIoPe21}. This is achieved by showing a cocycle superrigidty result for Bernoulli shifts of non-properly proximal groups. We also obtain that Bernoulli shifts of countable, nonamenable, i.c.c., exact, non-properly proximal groups are OE-superrigid.
Equivariant $KK$-theory of Bernoulli shifts on $C^*$-algebras with approximately inner flip
本文基于 Enders--Schemeitat--Tikuisis 分类，证明了 UCT 类中具有近似内翻转的可分 $C^*$-代数是 $K$-理论自吸收的当且仅当其 Bernoulli 平移 $KK^G$-等价于平凡作用，并推广了相关 $K$-理论公式。
Namely, we prove that if $G$ is a countable, discrete group and $f\in M_{n}(\Z(G))$ is invertible on $\ell^{2}(G)^{\oplus n},$ but $f$ is not invertible in $M_{n}(\Z(G))$, then the measure-preserving action of $G$ on $X_{f}$ equipped with the Haar measure is weakly equivalent to a Bernoulli action. We shall in fact prove this weak equivalence in the case that $f$ has a "formal inverse in $\ell^{2}$".
We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.
On the KMS states for the Bernoulli shift
本文研究Bernoulli移位交叉积上由单坐标势确定的流，证明对任意非零β存在II_∞型极值β-KMS态，并在势有理相关时确定III_λ型极值态存在的λ范围。
We determine the Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (\{0,1\},μ_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $μ_g$. We prove in particular that the action is never of type II$_\infty$ if $G$ is abelian and not locally finite, answering Krengel's question for $G = \mathbb{Z}$. When $G$ is locally finite, we prove that type II$_\infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$…
We consider II$_1$ factors of the form $M=\bar{\bigotimes}_{G}N\rtimes G$, where either i) $N$ is a non-hyperfinite II$_1$ factor and $G$ is an ICC amenable group or ii) $N$ is a weakly rigid II$_1$ factor and $G$ is ICC group and where $G$ acts on $\bar{\bigotimes}_{G}N$ by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting are isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.
Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},μ_g)$ with two base points. When $ G $ is exact, under a certain finiteness assumption on the measures $\{μ_g\}_{g\in G }$, we construct a boundary for the Bernoulli crossed product C$^*$-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigi…
We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.
We prove that for a weakly mixing algebraic action $σ: G\curvearrowright(X,ν)$, the $n$-cohomology group $H^n(G\curvearrowright X; \mathbb{T})$, after quotienting out the natural subgroup $H^n(G,\mathbb{T})$, contains $H^n(G,\widehat{X})$ as a natural subgroup for $n=1$. If we further assume the diagonal actions $σ^2, σ^4$ are $\mathbb{T}$-cocycle superrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, then the above also holds true for $n=2$. Applying it for principal algebraic actions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ is torsion free as an abelian group when $G$…
Previous work showed that all Bernoulli shifts over a free group are orbit-equivalent. This result is strengthened here by replacing Bernoulli shifts with the wider class of properly ergodic countable state Markov chains over a free group. A list of related open problems is provided.
We prove that any isomorphism $θ:M_0\simeq M$ of group measure space II$_1$ factors, $M_0=L^\infty(X_0, μ_0) \rtimes_{σ_0} G_0$, $M=L^\infty(X, μ) \rtimes_σ G$, with $G_0$ containing infinite normal subgroups with the relative property (T) of Kazhdan-Margulis (i.e. $G_0$ {\it w-rigid}) and $G$ an ICC group acting by Bernoulli shifts $σ$, essentially comes from an isomorphism of probability spaces which conjugates the actions. Moreover, any isomorphism $θ$ of $M_0$ onto a ``corner'' $pMp$ of $M$, for $p\in M$ an idempotent, forces $p=1$. In particular, all group measure space factors associated…
Bernoulli actions of type III$_0$ with prescribed associated flow
该研究针对群作用与算子代数交叉问题，证明了关联流的无穷可分结构性质，构造了指定关联流的伯努利作用与Poisson悬浮作用，还得到单射因子的张量积结构结论。
We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an algebraic action of $G$ on $X,$ there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is $G\curvearrowright X/Y$-ergodic "in the presence of $G\curvearrowright X$". We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal …
A countable group Γis called shift-minimal if every non-trivial measure preserving action of Γweakly contained in the Bernoulli shift of Γon ([0,1]^Γ,λ^Γ) is free. We show that any group Γwhose reduced C^*-algebra admits a unique tracial state is shift-minimal, and that any group Γadmitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that Γ/N is shift-minimal. Any shift-minimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shift-minimal groups. We also ex…
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