On C*-algebras associated to right LCM semigroups

TL;DR

This paper explores the structure of C*-algebras linked to right LCM semigroups, establishing conditions for their uniqueness, simplicity, and pure infiniteness, and analyzing the interplay between semigroup units and right ideals.

Contribution

It introduces a detailed analysis of C*-algebras from right LCM semigroups, highlighting conditions for uniqueness, simplicity, and pure infiniteness, based on algebraic properties.

Findings

Uniqueness results for full semigroup C*-algebras.

Identification of conditions for pure infiniteness and simplicity.

Analysis of the interaction between units and right ideals.

Abstract

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.