Expanders, exact crossed products, and the Baum–Connes conjecture

Abstract

We reformulate the Baum–Connes conjecture with coefficients by introducing a new crossed product functor for $C^{\ast }$-algebras. All confirming examples for the original Baum–Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum–Connes conjecture become confirming examples for our reformulated conjecture.