C*-algebras from partial isometric representations of LCM semigroups

TL;DR

This paper introduces a new way to construct C*-algebras from cancellative semigroups using partial isometric representations, with detailed analysis for LCM semigroups, including applications to free semigroups and self-similar groups.

Contribution

It generalizes existing constructions by providing a novel approach for LCM semigroups and connects these algebras to inverse semigroup and groupoid algebras.

Findings

Realization of algebras as inverse semigroup algebras

Realization of algebras as groupoid algebras

Application to free semigroups and Zappa-Szép products

Abstract

We give a new construction of a C*-algebra from a cancellative semigroup $P$ via partial isometric representations, generalising the construction from the second named author's thesis. We then study our construction in detail for the special case when $P$ is an LCM semigroup. In this case we realize our algebras as inverse semigroup algebras and groupoid algebras, and apply our construction to free semigroups and Zappa-Sz\'ep products associated to self-similar groups.