Inverse semigroup equivariant $KK$-theory and $C^*$-extensions

TL;DR

This paper extends Kasparov's classical isomorphism between KK-theory and extension groups to the inverse semigroup equivariant setting, broadening the scope of the theory.

Contribution

It generalizes the KK-theory and extension group isomorphism to inverse semigroup actions, adapting Thomsen's proof for this broader context.

Findings

Established isomorphism for inverse semigroup equivariant KK-theory and extension groups.

Extended known results from group actions to inverse semigroup actions.

Provided technical adaptation of Thomsen's proof for the inverse semigroup setting.

Abstract

In this note we extend the classical result by G. G. Kasparov that the Kasparov groups $KK_1(A,B)$ can be identified with the extension groups $\mbox{Ext}(A,B)$ to the inverse semigroup equivariant setting. More precisely, we show that $KK_G^1(A,B) \cong \mbox{Ext}_G(A \otimes {\cal K}_G,B \otimes {\cal K}_G)$ for every countable, $E$-continuous inverse semigroup $G$. For locally compact second countable groups $G$ this was proved by K. Thomsen, and technically this note presents an adaption of his proof.