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 $\Gamma < G= G_1 \times \dots \times G_d$ 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 $\mathcal{N}\rtimes\Gamma$ and $(L^\infty(G/P,\nu_P)\overline{\otimes}\mathcal{N})\rtimes\Gamma$ is again a crossed product of the form $(L^\infty(G/Q,\nu_Q)\overline{\otimes}\mathcal{N})\rtimes\Gamma$.