We introduce the first examples of groups \(G\) with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, \(\mathcal{L}(G)\). Specifically, assume ...that \(G=A\times W\), where \(A\) is an infinite abelian group and \(W\) is an ICC wreath-like product group CIOS22a; AMCOS23 with property (T) and trivial abelianization. Then whenever \(H\) is an \emph{arbitrary} group such that \(\mathcal{L}(G)\) is \(\ast\)-isomorphic to \(\mathcal L(H)\), via an \emph{arbitrary} \(\ast\)-isomorphism preserving the canonical traces, it must be the case that \(H= B \times H_0\) where \(B\) is infinite abelian and \(H_0\) is isomorphic to \(W\). Moreover, we completely describe the \(\ast\)-isomorphism between \(\mathcal L(G)\) and \(\mathcal L(H)\). This yields new applications to the classification of group C\(^*\)-algebras, including examples of non-amenable groups which are recoverable from their reduced C\(^*\)-algebras but not from their von Neumann algebras.
The note establishes a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Key words and phrases: affine action, mean ergodic ...theorem, groups of polynomial growth.
We use deformation-rigidity theory in the von Neumann algebra framework to study probability measure preserving actions by wreath product groups. In particular, we single out large families of wreath ...product groups satisfying various types of orbit equivalence (OE) rigidity. For instance, we show that whenever H, K, Γ, Λ are icc, property (T) groups such that H≀Γ and K≀Λ admit stably orbit equivalent action σ and ρ such that σ|Γ, ρ|Λ, σ|HΓ, and ρ|KΛ are ergodic, then automatically σΓ is stably orbit equivalent to ρΛ and σ|HΓ is stably orbit equivalent to ρ|KΛ. Rigidity results for von Neumann algebras arising from certain actions of such groups (i.e. W⁎-rigidity results) are also obtained.
In this paper we introduce a new family of icc groups $\Gamma$ which satisfy
the following product rigidity phenomenon, discovered in DHI16 (see also
dSP17): all tensor product decompositions of the ...II$_1$ factor $L(\Gamma)$
arise only from the canonical direct product decompositions of the underlying
group $\Gamma$. Our groups are assembled from certain HNN-extensions and
amalgamated free products and include many remarkable groups studied throughout
mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes
groups, Higman group, various integral two-dimensional Cremona groups, etc. As
a consequence, we obtain several new examples of groups that give rise to prime
factors.
In \cite{CDD22} we investigated the structure of \(\ast\)-isomorphisms between von Neumann algebras \(L(\Gamma)\) associated with graph product groups \(\Gamma\) of flower-shaped graphs and property ...(T) wreath-like product vertex groups as in \cite{CIOS21}. In this follow-up we continue the structural study of these algebras by establishing that these graph product groups \(\Gamma\) are entirely recognizable from the category of all von Neumann algebras arising from an arbitrary non-trivial graph product group with infinite vertex groups. A sharper \(C^*\)-algebraic version of this statement is also obtained. In the process of proving these results we also extend the main \(W^*\)-superrigidity result from \cite{CIOS21} to direct products of property (T) wreath-like product groups.
Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group \(\Gamma\) its von Neumann algebra ...\(L(\Gamma)\) satisfies the so-called ISR property: \emph{any von Neumann subalgebra \(N\subseteq L(\Gamma)\) that is normalized by all group elements in \(\Gamma\) is of the form \(N= L(\Sigma)\) for a normal subgroup \(\Sigma \lhd \Gamma\).} In particular, this applies to all groups \(\Gamma\) in each of the following classes: all icc (relatively) hyperbolic groups, most mapping class groups of surfaces, all outer automorphisms of free groups with at least three generators, most graph product groups arising from simple graphs without visual splitting, etc. This result answers positively an open question of Amrutam and Jiang from \cite{AJ22}. In the second part of the paper we obtain similar results for factors associated with groups that admit nontrivial (quasi)cohomology valued into various natural representations. In particular, we establish the ISR property for all icc, nonamenable groups that have positive first \(L^2\)-Betti number and contain an infinite amenable subgroup.
In this short note we classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces.
A group \(G\) is called \(W^*\)-superrigid (resp. \(C^*\)-superrigid) if it is completely recognizable from its von Neumann algebra \(L(G)\) (resp. reduced \(C^*\)-algebra \(C_r^*(G)\)). Developing ...new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of \(W^*\)-superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in \(C^*\)-algebra theory including additional examples of \(C^*\)-superrigid groups and explicit computations of symmetries of reduced group \(C^*\)-algebras.
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