A Green--Julg isomorphism for inverse semigroups

TL;DR

This paper extends the Green--Julg isomorphism, a fundamental result in operator algebras, from finite groups to finite inverse semigroups, establishing a new isomorphism between equivariant KK-theory and crossed product K-theory.

Contribution

It introduces a Green--Julg isomorphism for finite inverse semigroups, broadening the scope of the classical result from group actions to inverse semigroup actions.

Findings

Established an isomorphism between KK^S(ℂ,A) and K(A ⋊ S) for finite inverse semigroups.

Extended the classical Green--Julg isomorphism from groups to inverse semigroups.

Provides a new tool for analyzing crossed products by inverse semigroup actions.

Abstract

For every finite unital inverse semigroup $S$ and $S$-$C^*$-algebra $A$ we establish an isomorphism between $KK^S(\mathbb{C},A)$ and $K(A \rtimes S)$. This extends the classical Green--Julg isomorphism from finite groups to finite inverse semigroups.