Kk-theoretic Duality for Proper Twisted Actions

Let the discrete group G act properly and isometrically on the Riemannian manifold X. Let C0(X, δ) be the section algebra of a smooth locally trivial G-equivariant bundle of elementary C∗-algebras representing an element δ of the Brauer group BrG(X). Then C0(X, δ ) ⋊ G is KK-theoretically Poincaré dual to ( C0(X, δ)⊗̂C0(X)Cτ (X) ) ⋊ G, where δ is the inverse of δ in the Brauer group. We deduce this from a strengthening of Kasparov’s duality theorem RKK(X;A,B) ∼= KK(Cτ (X)⊗̂A,B). As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifolds M and Poincaré dualities for twisted group algebras C∗(G,ω) if G satisfies some additional properties related to the Dirac dual-Dirac method for the Baum-Connes conjecture.