The isomorphism problem for semigroup C*-algebras of right-angled Artin monoids

TL;DR

This paper completely characterizes when semigroup C*-algebras of right-angled Artin monoids are isomorphic, using graph properties and K-theory, and identifies which are isomorphic to graph algebras, revealing many are semiprojective.

Contribution

It provides a full classification of isomorphism conditions for these C*-algebras based on graph properties and K-theory, answering key open questions.

Findings

Complete classification of isomorphism conditions

Identification of when these algebras are graph algebras

Many of the studied C*-algebras are semiprojective

Abstract

Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca. In the paper at hand, we are able to present the complete answer to their question of when such C*-algebras are isomorphic. The answer to this question is presented both in terms of properties of the graph defining the Artin monoids as well as in terms of classification by K-theory, and is obtained using recent results from classification of non-simple C*-algebras. Moreover, we are able to answer another natural question: Which of these semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph algebras? We give a complete answer, and note the consequence that many of the C*-algebras under study are semiprojective.