C*-algebras of labelled spaces and their diagonal C*-subalgebras

TL;DR

This paper constructs a representation linking inverse semigroups and C*-algebras of labelled spaces, demonstrating a homeomorphism between the spectrum of a diagonal subalgebra and the inverse semigroup's tight spectrum.

Contribution

It introduces a new representation of labelled space C*-algebras inspired by path operations, connecting inverse semigroup theory with operator algebras.

Findings

Non-zero inverse semigroup elements correspond to non-zero C*-algebra elements

Spectrum of diagonal subalgebra is homeomorphic to the inverse semigroup's tight spectrum

Representation method reflects path cutting and gluing operations

Abstract

Motivated by Exel's inverse semigroup approach to combinatorial C*-algebras, in a previous work the authors defined an inverse semigroup associated with a labelled space. We construct a representation of the C*-algebra of a labelled space, inspired by how one might cut or glue labelled paths together, that proves that non-zero elements in the inverse semigroup correspond to non-zero elements in the C*-algebra. We also show that the spectrum of its diagonal C*-subalgebra is homeomorphic to the tight spectrum of the inverse semigroup associated with the labelled space.