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 ) ⊗ ¯ Γ ⋊ Γ is solid relatively to L ( Γ ) . In particular, if L ( Γ ) is solid then M is solid and if Γ is non-amenable and A 0 ≠ C then M is a full prime factor. This gives many new examples of solid or prime type III factors. Following Chifan and Ioana, we also obtain the first examples of solid non-amenable type III equivalence relations.