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

TL;DR

This paper proves a non-commutative version of the Intermediate Factor Theorem for crossed products involving product lattices in higher rank semisimple groups, characterizing intermediate von Neumann algebras.

Contribution

It extends the Intermediate Factor Theorem to a non-commutative setting for tensor products of lattices in product groups, providing a classification of intermediate von Neumann algebras.

Findings

Every intermediate von Neumann algebra is a crossed product of a specific form.

The theorem applies to irreducible lattices in higher rank semisimple algebraic groups.

The result generalizes classical commutative factor theorems to non-commutative operator algebras.

Abstract

We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $\Gamma < G= G_1 \times \dots \times G_d$ 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 $(L^\infty(G/Q,\nu_Q)\overline{\otimes}\mathcal{N})\rtimes\Gamma$.