Solidity of type III Bernoulli crossed products

TL;DR

This paper extends solidity results to type III Bernoulli crossed products, showing they are solid relative to group von Neumann algebras, leading to new examples of prime and solid type III factors and equivalence relations.

Contribution

It generalizes previous theorems to include type III von Neumann algebras, establishing solidity and primeness of associated Bernoulli crossed products.

Findings

Bernoulli crossed products are solid relative to group von Neumann algebras.

If the group von Neumann algebra is solid, the crossed product is also solid.

Non-amenable groups with non-trivial base algebras produce prime type III factors.

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 $\varphi_0$ and any discrete group $\Gamma$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,\varphi_0)^{\mathbin{\bar{\otimes}}\Gamma}\rtimes \Gamma$ 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 $A_0 \neq \mathbb{C}$ 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.