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 $\Gamma <G=G_1\times \cdots \times G_d$ 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 $\mathcal{N}⋊\Gamma$ and $(L^{\infty }(G/P,{\nu }_P)\overline{\otimes }\mathcal{N})⋊\Gamma$ is again a crossed product of the form $(L^{\infty }(G/Q,{\nu }_Q)\overline{\otimes }\mathcal{N})⋊\Gamma$.

14 pages; comments are welcome