A new uniqueness theorem for the tight C*-algebra of an inverse semigroup

TL;DR

This paper establishes a new uniqueness theorem for tight C*-algebras of inverse semigroups, extending previous results for groupoid C*-algebras, and applies it to clarify the structure of certain subshift-related C*-algebras.

Contribution

It generalizes the uniqueness theorem to inverse semigroup C*-algebras and clarifies the identification of the tight C*-algebra associated with subshifts.

Findings

Injectivity of *-homomorphisms characterized by core submonoid

Clarification of the identity of a specific tight C*-algebra

Extension of uniqueness theorems to inverse semigroup context

Abstract

We prove a new uniqueness theorem for the tight C*-algebras of an inverse semigroup by generalizing the uniqueness theorem given for \'etale groupoid C*-algebras by Brown, Nagy, Reznikoff, Sims, and Williams. We use this to show that in the nuclear and Hausdorff case, a *-homomorphism from the boundary quotient C*-algebra of a right LCM monoid is injective if and only if it is injective on the subalgebra generated by the core submonoid. We also use our result to clarify the identity of the tight C*-algebra of an inverse semigroup we previously associated to a subshift and erroneously identified as the Carlsen-Matsumoto algebra.