Connes’ rigidity conjecture for higher-rank lattices

Establish that for semisimple connected real Lie groups G1 and G2 with trivial centers, no compact factors, and real rank at least 2, and irreducible lattices Γ1 < G1 and Γ2 < G2, the equality of group von Neumann algebras L(Γ1) = L(Γ2) implies G1 = G2 and, consequently, rk_R(G1) = rk_R(G2).

Background

Connes’ rigidity conjecture is a central open problem in the classification program for group von Neumann algebras associated with lattices in higher-rank semisimple Lie groups. For a countable discrete group Γ, the group von Neumann algebra L(Γ) encodes operator-algebraic information about Γ; the conjecture posits that, for irreducible lattices in connected, centerless, noncompact semisimple real Lie groups of real rank at least 2, this operator-algebraic invariant fully determines the ambient Lie group.

This paper develops an operator-algebraic characterization of noncommutative Furstenberg–Poisson boundaries and uses it, together with prior results, to derive new evidence for the conjecture. Specifically, Theorem C shows that under an additional normality assumption on a ucp map between boundary algebras, one can conclude equality of real ranks, thereby supporting the rigidity phenomena predicted by Connes’ conjecture.