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*-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.

53 pages. Some (relatively minor) changes and corrections

References in corpus (6)

• Small cancellation theory and Burnside problem
• Exotic group C*-algebras in noncommutative duality
• Imprimitivity theorems for weakly proper actions of locally compact groups
• The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space
• Ghostbusting and property A
• Free group $C^{\ast }$-algebras associated with ${\ell }_p$

Cited by in corpus (16)

• Novikov's Conjecture
• Coaction functors
• Exotic group $C^{\ast }$-algebras of simple Lie groups with real rank one
• Straightening warped cones
• Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model
• Warped cones over profinite completions
• Coaction functors, II
• Tensor-product coaction functors
• Slant products on the Higson-Roe exact sequence
• The maximal injective crossed product
• Ordered invariant ideals of Fourier-Stieltjes algebras
• Crossed products of operator spaces
• R-coactions on $C^{\ast }$-algebras
• Group $C^{\ast }$-algebras of locally compact groups acting on trees
• Tensor $D$ coaction functors
• On the Baum--Connes conjecture with coefficients for linear algebraic groups