Non-commutative Intermediate Factor theorem associated with $W^*$-dynamics of product groups

Let $G = G_{1} \times G_{2}$ be a product of two locally compact, second countable groups and $\mu \in \mathrm{Prob}(G)$ be of the form $\mu = \mu_{1} \times \mu_{2}$, where $\mu_{i} \in \mathrm{Prob}(G_{i})$. Let $(B,\nu_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\mathcal{M}$ with $$\mathcal{N} \subseteq \mathcal{M} \subseteq \mathcal{N} \,\bar{\otimes}\, L^{\infty}(B,\nu)$$ splits as a tensor product of the form $\mathcal{N}\bar{\otimes}L^{\infty}(C,\nu_C)$, where $(C,\nu_C)$ is a $(G,\mu)$-boundary. Here, $\mathcal{N}$ is a tracial von Neumann algebra on which $G$ acts trace-preservingly. This generalizes the Intermediate Factor Theorem proved by Bader--Shalom (\cite[Theorem~1.9]{BS06}) in the measurable setup. In addition, we give various other examples of the splitting phenomenon associated with $W^{*}$-dynamics. We also show that certain assumptions are necessary for the intermediate algebras to split, and ideals in the ambient tensor product algebra obstruct the splitting phenomenon. We also use the Master theorem from \cite{glasner2023intermediate} to resolve the second part of \cite[Problem~5.2]{jiangskalski} in the affirmative.