kk-Theory for Banach algebras II: Equivariance and Green–Julg type theorems

Abstract

We extend the definition of the bivariant K-theory kk ban from plain Banach algebras to Banach algebras equipped with an action of a locally compact Hausdorff group G. We also define a natural transformation from Lafforgue's theory KK ban G into the new equivariant theory, overcoming some technical difficulties that are particular to the equivariant case. The categorical framework allows us to systematically define a descent homomorphism and to prove a Green-Julg theorem, a dual version of it and a generalised version that involves the action of a proper G-space. We also include a naïve Poncaré duality theorem.