E-theory and KK-theory for groups which act properly and isometrically on Hilbert space

Abstract

A good deal of research in C∗-algebra K -theory in recent years has been devoted to the Baum-Connes conjecture [3], which proposes a formula for the K -theory of group C∗-algebras that blends group homology with the representation theory of compact subgroups. The conjecture has brought C∗algebra theory into close contact with manifold theory through its obvious similarity to the Borel conjecture of surgery theory [9,31] and its links with the theory of positive scalar curvature [27]. In addition there are points of contact with harmonic analysis, particularly the tempered representation theory of semisimple groups, although the proper relation between the Baum-Connes conjecture and representation theory is not well understood. The conjecture is most easily formulated for groups which are discrete and torsion-free. For such a group G there is a natural homomorphism