Invariant von Neumann subalgebras of crossed products and tensor products

Abstract

Let $Γ$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,τ_Q)$ in a trace-preserving way. We study invariant von Neumann subalgebras of $Q\rtimesΓ$ and of $L(Γ)\bar\otimes Q$, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto $Q$ is scalar is necessarily contained in $L(Γ)$. Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with $Q$. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of $\mathbb{F}_n$ for every $2\leq n\leq\infty$, as well as non-prime examples with explicitly described factors.