The noncommutative factor theorem for lattices in product groups

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 $\mathrm{\Gamma }<G$ in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras $\mathrm{L}(\mathrm{\Gamma })\subset M\subset \mathrm{L}(\mathrm{\Gamma }\curvearrowright G\mathrm{/}P)$ sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.