The failure of exactness (in the sense of C*-algebra theory) is at the heart of counterexamples to the Baum-Connes conjecture. To account for this, we reformulate the conjecture by introducing a new crossed product functor. For exact groups the conjecture is unchanged, and all confirming examples remain confirming examples for the reformulated conjecture. However, some of the known expander-based counterexamples to the original conjecture become confirming examples for the reformulated conjecture.

This is joint work with Paul Baum and Rufus Willett.