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

• DOI: 10.1016/j.jmaa.2017.06.074
• Publication Date: 2017-07-04T16:46:25Z
• Authors: Giuliano Boava; Gilles G. de Castro; Fernando de L. Mo...
• Keywords: Inverse semigroup; Inverse element; Zero (linguistics); Bicyclic semigroup; Representation

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.<br/>30 pages<br/>