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We prove that if A is a non-separable abelian tracial von Neuman algebra then its free powers A∗n,2≤n≤∞, are mutually non-isomorphic and with trivial fundamental group, , whenever 2≤n<∞. This settles the non-separable version of the free group factor problem. [ABSTRACT FROM AUTHOR]

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees. [ABSTRACT FROM AUTHOR]

We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in [ 5 ]. This property has applications to strong ergodicity, the Banach–Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group |$\operatorname{Isom}\left (\mathbb{R}^{d}\right)$| of orientation-preserving isometries of the Euclidean space |$\mathbb{R}^{d}$|⁠ , for d ≥ 3. We prove that the translation action of a countable dense subgroup Γ on Isom |$\left (\mathbb R^{d}\right)$| has local spectral gap, whenever the translation action of the rotation projection of Γ on SO(d) has spectral gap. Our proof relies on the amenability of |$\operatorname{Isom}\left (\mathbb{R}^{d}\right)$| and on work of Lindenstrauss and Varjú [ 12 ]. [ABSTRACT FROM AUTHOR]

We show that Shlyakhtenko's free Araki-Woods factors are strongly solid, meaning that for any diffuse amenable von Neumann subalgebra that is the range of a normal conditional expectation, the normalizer remains amenable. This provides the first class of nonamenable strongly solid type III factors. [ABSTRACT FROM AUTHOR]

We investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebrasM =M1 *BM2.Our main result states that, under natural analytic assumptions, any amenable subalgebra of M that has a large intersection with M1 is actually contained in M1. The proof does not rely on Popa's asymptotic orthogonality property but on the study of nonnormal conditional expectations. [ABSTRACT FROM AUTHOR]

We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $$\Gamma \curvearrowright G$$ , whenever $$\Gamma $$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83-121, 2008; J Eur Math Soc (JEMS) 14:1455-1511, 2012), and Benoist and de Saxcé (Invent Math 205:337-361, 2016). We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $$\Gamma $$ -invariant finitely additive measure defined on all bounded measurable subsets of G. [ABSTRACT FROM AUTHOR]

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms of Λ-invariant measures on some compact Γ-space. The strategy of proof is different from Popa's approach to maximal amenability via central sequences (Adv Math 50:27-48, ), and relies on elementary computations in a crossed-product C-algebra. [ABSTRACT FROM AUTHOR]

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $. [ABSTRACT FROM AUTHOR]

Abstract: We generalize -superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action is -superrigid. More precisely, if is another free ergodic action such that the crossed-product von Neumann algebras are isomorphic , then the actions are conjugate. We prove a similar statement whenever is a non-amenable ICC product of two infinite groups. [Copyright &y& Elsevier]

Abstract: We investigate Gaussian actions through the study of their crossed-product von Neumann algebra. The motivational result is Chifan and Ioanaʼs ergodic decomposition theorem for Bernoulli actions (Chifan and Ioana, 2010 ) that we generalize to Gaussian actions (Theorem A). We also give general structural results (Theorems 3.4 and 3.8) that allow us to get a more accurate result at the level of von Neumann algebras. More precisely, for a large class of Gaussian actions , we show that any subfactor N of containing is either hyperfinite or is non-Gamma and prime. At the end of the article, we show a similar result for Bogoliubov actions. [Copyright &y& Elsevier]