Non-commutative Factor theorem for tensor products of lattices in product groups

We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $\mathrm{\Gamma }<G=G_1\times \cdots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G\curvearrowright (\mathcal{N},\tau )$, we show that every intermediate von Neumann algebra between $\mathcal{N}⋊\mathrm{\Gamma }$ and $(L^{\mathrm{\infty }}(G\mathrm{/}P,{\nu }_P)\bar{\otimes }\mathcal{N})⋊\mathrm{\Gamma }$ is again a crossed product of the form $(L^{\mathrm{\infty }}(G\mathrm{/}Q,{\nu }_Q)\bar{\otimes }\mathcal{N})⋊\mathrm{\Gamma }$.