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

Abstract: We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $Γ&lt; G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G \curvearrowright (\mathcal{N}, τ)$, we show that every intermediate von Neumann algebra between $\mathcal{N}\rtimesΓ$ and $$(L<sup>\infty(G/P,ν_P)\overline{\otimes}\mathcal{N})\rtimesΓ$$ is again a crossed product of the form $$(L<sup>\infty(G/Q,ν_Q)\overline{\otimes}\mathcal{N})\rtimesΓ$$.