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

Let $G$ denote a countable inverse semigroup. We construct a kind of a Baum–Connes map $K(\tilde{A}⋊G)\to K(A⋊G)$ 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”.