The universal property of inverse semigroup equivariant $KK$-theory

TL;DR

This paper extends Higson's universal property of KK-theory to the inverse semigroup equivariant setting, broadening the theoretical framework for functors from $C^*$-algebras.

Contribution

It generalizes Higson's factorization result to inverse semigroup equivariant KK-theory, adapting Thomsen's group equivariant approach.

Findings

Universal property established for inverse semigroup equivariant KK-theory

Functor factorization through KK-theory proven in this setting

Extension of Higson's theorem to a broader algebraic context

Abstract

Higson proved that every homotopy invariant, stable and split exact functor from the category of $C^*$-algebras to an additive category factors through Kasparov's $KK$-theory. By adapting a group equivariant generalization of this result by Thomsen, we generalize Higson's result to the inverse semigroup equivariant setting.