The K-theory of some reduced inverse semigroup C*-algebras

TL;DR

This paper computes the K-theory of reduced inverse semigroup C*-algebras by relating them to crossed products, extending existing results to new classes like graph inverse semigroups and tilings.

Contribution

It establishes a Morita equivalence between inverse semigroup C*-algebras and certain crossed products, enabling K-theory calculations for new inverse semigroup classes.

Findings

K-theory description for inverse semigroup C*-algebras

Morita equivalence with crossed products

Application to graph inverse semigroups and tilings

Abstract

We use a recent result by Cuntz, Echterhoff and Li about the K-theory of certain reduced C*-crossed products to describe the K-theory of C*_r(S) when S is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that C*_r(S) is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.