A note on a certain Baum--Connes map for inverse semigroups

TL;DR

This paper develops a categorical approach to define a Baum--Connes map for countable inverse semigroups, extending previous work for groups, and introduces fibered G-algebras as coefficients.

Contribution

It constructs a Baum--Connes map for inverse semigroups using localization of triangulated categories, extending the framework from groups to inverse semigroups.

Findings

Constructed a Baum--Connes map for inverse semigroups.

Extended categorical localization techniques to inverse semigroups.

Introduced fibered G-algebras as a class of coefficient algebras.

Abstract

Let $G$ denote a countable inverse semigroup. We construct a kind of a Baum--Connes map $K(\tilde A \rtimes G) \rightarrow K(A \rtimes G)$ by a categorial approach via localization of triangulated categories, developed by R. Meyer and R. Nest for groups $G$. We allow the coefficient algebras $A$ to be in a special class of algebras called fibered $G$-algebras. This note continues and fixes our preprint "Attempts to define a Baum--Connes map via localization of categories for inverse semigroups".