The noncommutative factor theorem for lattices in product groups

Abstract

We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices Γ<G in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras L(Γ)⊂M⊂L(Γ↷G/P) sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.