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Solidity of type III Bernoulli crossed products

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 $φ_0$ and any discrete group $Γ$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,φ_0)^{\mathbin{\bar{\otimes}}Γ}\rtimes Γ$ is solid relatively to $\mathcal{L}(Γ)$. In particular, if $\mathcal{L}(Γ)$ is solid then $M$ is solid and if $Γ$ 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.