Solidity of type III Bernoulli crossed products

Abstract

We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state ${\phi }_0$ and any discrete group $\Gamma$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,{\phi }_0{)}^{\bar{\otimes }\Gamma }⋊\Gamma$ is solid relatively to $\mathcal{L}(\Gamma )$. In particular, if $\mathcal{L}(\Gamma )$ is solid then $M$ is solid and if $\Gamma$ is non-amenable and $A_0\ne \mathbb{C}$ then $M$ is a full prime factor. This gives many new examples of solid or prime type $\mathrm{III}$ factors. Following Chifan and Ioana, we also obtain the first examples of solid non-amenable type $\mathrm{III}$ equivalence relations.

18 pages

References in corpus (6)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Unique prime factorization and bicentralizer problem for a class of type III factors
• Asymptotic structure of free product von Neumann algebras
• Classification of type III Bernoulli crossed products
• Rigidity of free product von Neumann algebras
• Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence

Cited by in corpus (2)

• Boundary and rigidity of nonsingular Bernoulli actions
• Classification results for nonsingular Bernoulli crossed products