Character rigidity for lattices in higher-rank groups

We show that if Γ is an irreducible lattice in a higher rank center-free semi-simple Lie group with no compact factors and having property (T) of Kazhdan, then Γ is operator algebraic superrigid, i.e., any unitary representation of Γ which generates a II1 factor extends to a homomorphism of the group von Neumann algebra LΓ. This generalizes results of Margulis, and Stuck and Zimmer, and answers in the affirmative a conjecture of Connes. We also show a general operator algebraic superrigidity result for irreducible lattices in products of property (T) groups, generalizing results of Bader and Shalom, and Creutz.