Semigroup *-Algebras Arising from Graphs of Monoids

Abstract

Abstract We study groupoids and semigroup $C^{\ast }$-algebras arising from graphs of monoids, in the setting of right LCM monoids. First, we establish a general criterion when a graph of monoids gives rise to a submonoid of the fundamental group that is right LCM. Moreover, we carry out a detailed analysis of structural properties of semigroup $C^{\ast }$-algebras arising from graphs of monoids, including closed invariant subspaces and topological freeness of the groupoids, as well as ideal structure, nuclearity, and K-theory of the semigroup $C^{\ast }$-algebras. As an application, we construct families of pairwise nonconjugate Cartan subalgebras in every UCT Kirchberg algebra.