Invariant von Neumann subalgebras of crossed products and tensor products

Let $\mathrm{\Gamma }$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,{\tau }_Q)$ in a trace-preserving way. We study invariant von Neumann subalgebras of $Q⋊\mathrm{\Gamma }$ and of $L(\mathrm{\Gamma })\bar{\otimes }Q$, 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 $L(\mathrm{\Gamma })$. 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 $2\le n\le \mathrm{\infty }$, as well as non-prime examples with explicitly described factors.