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

TL;DR

This paper extends bivariant K-theory for Banach algebras to include group actions, establishing a framework for equivariant theories, descent homomorphisms, and Green-Julg type theorems.

Contribution

It introduces an equivariant version of $kk^{ban}$ for Banach algebras with group actions and proves related theorems, including a descent homomorphism and Green-Julg theorems.

Findings

Defined a new equivariant $kk^{ban}_G$ theory.

Proved a Green-Julg theorem and its dual.

Established a descent homomorphism and a Poincaré duality theorem.

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 na\"{i}ve Poncar\'{e} duality theorem.