$kk$-Theory for Banach Algebras II: Equivariance and Green-Julg type theorems

Abstract

We extend the definition of the bivariant $K$-theory $k{k}^{ban}$ 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 $K{K}_G^{ban}$ 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.

38 pages

References in corpus (1)

• A generalised Green-Julg theorem for proper groupoids and Banach algebras