Bass-Serre rigidity results in von Neumann algebras

Abstract

We obtain new Bass-Serre type rigidity results for $\mathrm{I}{\mathrm{I}}_1$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras ${\mathcal{M}}_1{\ast }_B{\mathcal{M}}_2$ over an abelian von Neumann algebra $B$, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type $\mathrm{I}{\mathrm{I}}_1$ and in the type $\mathrm{III}$ case, new examples of prime factors.

27 pages.

References in corpus (3)

• Ergodic Subequivalence Relations Induced by a Bernoulli Action
• Construction of type $\mathrm{I}{\mathrm{I}}_1$ factors with prescribed countable fundamental group
• Free products, Orbit Equivalence and Measure Equivalence Rigidity

Cited by in corpus (16)

• Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors
• Group measure space decomposition of II_1 factors and W*-superrigidity
• Factoriality, type classification and fullness for free product von Neumann algebras
• Amalgamated free product type III factors with at most one Cartan subalgebra
• Structure of $\mathrm{I}{\mathrm{I}}_1$ factors arising from free Bogoljubov actions of arbitrary groups
• Asymptotic structure of free product von Neumann algebras
• Rigidity of free product von Neumann algebras
• A Unique Prime Decomposition Result for Wreath Product Factors
• Primeness results for von Neumann algebras associated with surface braid groups
• W*-rigidity paradigms for embeddings of II${}_1$ factors
• Some $OE$ and $W^{\ast }$-rigidity results for actions by wreath product groups
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups
• Asymptotic freeness in tracial ultraproducts
• On ergodic embeddings of factors
• Structure of extensions of free Araki-Woods factors
• Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras