We provide a fairly large family of amalgamated free product groups Γ=Γ1⁎ΣΓ2 whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that Γi is a ...product of two icc non-amenable bi-exact groups, and Σ is icc amenable with trivial one-sided commensurator in Γi, for every i=1,2. Then Γ satisfies the following rigidity property: any group Λ such that L(Λ) is isomorphic to L(Γ) admits an amalgamated free product decomposition Λ=Λ1⁎ΔΛ2 such that the inclusions L(Δ)⊆L(Λi) and L(Σ)⊆L(Γi) are isomorphic, for every i=1,2. This result significantly strengthens some of the previous Bass–Serre rigidity results for von Neumann algebras. As a corollary, we obtain the first examples of amalgamated free product groups which are W⁎-superrigid.
Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group Γ its von Neumann algebra L(Γ) ...satisfies the so-called ISR property: any von Neumann subalgebraN⊆L(Γ)that is normalized by all group elements in Γ is of the formN=L(Σ)for a normal subgroupΣ◁Γ. In particular, this applies to all groups Γ 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 2.
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 L2-Betti number and contain an infinite amenable subgroup.
A countable discrete 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 Cr⁎(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.
We prove that every separable tracial von Neumann algebra embeds into a II1 factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an ...analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. In addition, we obtain two new results concerning the structure of infinitely generic II1 factors. These results are obtained by using the class of wreath-like product groups introduced recently in 8.
In this paper we explore a generic notion of superrigidity for von Neumann algebras L(G) and reduced C⁎-algebras Cr⁎(G) associated with countable discrete groups G. This allows us to classify these ...algebras for various new classes of groups G from the realm of coinduced groups.
In this article we provide the first examples of property (T) $\mathrm{II}_1$ factors $\mathcal{N}$ with trivial fundamental group, $\mathcal{F}(\mathcal{N})=1$. Our examples arise as group factors ...$\mathcal{N}=\mathcal{L}(G)$ where $G$ belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in Geom. Dedicata 112 (2005), 183--196 and the ones introduced recently in Anal. PDE 16 (2023), 433--476 using the Belegradek-Osin Rips construction from Groups Geom. Dyn. 2 (2008), 1--12. In particular, our results provide a continuum of explicit pairwise nonisomorphic property (T) factors.
An exotic II1 factor without property Gamma Chifan, Ionuţ; Ioana, Adrian; Kunnawalkam Elayavalli, Srivatsav
Geometric and functional analysis,
2023/10, Volume:
33, Issue:
5
Journal Article
Peer reviewed
Open access
We introduce a new iterative amalgamated free product construction of II
1
factors, and use it to construct a separable II
1
factor which does not have property Gamma and is not elementarily ...equivalent to the free group factor
L
(
F
n
)
, for any
2
≤
n
≤
∞
. This provides the first explicit example of two non-elementarily equivalent II
1
factors without property Gamma. Moreover, our construction also provides the first explicit example of a II
1
factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory.
We study the relationship between the dynamics of the action α of a discrete group G on a von Neumann algebra M, and structural properties of the associated crossed product inclusion L(G)⊆M⋊αG, and ...its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion L(G)⊆M⋊αG such as the Haagerup Approximation Property, and is essential to capturing “almost periodic” behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial W⁎-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.
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