KK-theoretic duality for proper twisted actions

Abstract

Let the discrete group G act properly and isometrically on the Riemannian manifold X. Let C_0(X, \delta) be the section algebra of a smooth locally trivial G-equivariant bundle of elementary C*-algebras representing an element \delta of the Brauer group Br_G(X). Then C_0(X,\delta^{-1}) x G is KK-theoretically Poincare dual to (C_0(X,\delta)\otimes_{C_0(X)} C_\tau(X)) xG, where \delta^{-1} is the inverse of \delta in the Brauer group. We deduce this from a strengthening of Kasparov's duality theorem RKK^G(X; A,B) \cong KK^G(C_\tau(X)\otimes A, B). As applications we also obtain a version of the above Poincare duality with X replaced by a compact G-manifold M and for twisted group algebras C*(G,\omega) if G satisfies some additional properties related to the Dirac-dual Dirac method for the Baum-Connes conjecture.