Groupoid Models for the C*-Algebra of Labelled Spaces

Abstract

We define a groupoid from a labelled space and show that it is isomorphic to the tight groupoid arising from an inverse semigroup associated with the labelled space. We then define a local homeomorphism on the tight spectrum that is a generalization of the shift map for graphs, and show that the defined groupoid is isomorphic to the Renault-Deaconu groupoid for this local homeomorphism. Finally, we show that the C*-algebra of this groupoid is isomorphic to the C*-algebra of the labelled space as introduced by Bates and Pask.

Simplicity of inverse semigroup and étale groupoid algebras

“…(5) A matrix algebra over a regular ring is regular. (6) The center of a regular ring is regular. (7) Every semisimple Artinian ring is regular.…”

AF‐embeddable labeled graph C∗‐algebras

“…Remark After the first version of the present paper was posted, Boava, Castro and Mortani [5] constructed the groupoid model for a labeled space. Using their groupoid model, one sees that is isomorphic to a ‐algebra of a Hausdorff étale groupoid which can be shown to be ample and topologically principal.…”

Ultragraph algebras via labelled graph groupoids, with applications to generalized uniqueness theorems

“…Let (E, L, B) be a weakly-left resolving labelled space. We recall the description of the tight spectrum of E(S), as given in [5] with the correction pointed out in [7] and [17]. Let α ∈ L ≤∞ and let {F n } 0≤n≤|α| (understanding that 0 ≤ n ≤ |α| means 0 ≤ n < ∞ when α ∈ L ∞ ) be a family such that F n is a filter in B α 1,n for every n > 0 and F 0 is either a filter in B or F 0 = ∅.…”

“…Next, associate a topological groupoid to an arbitrary ultragraph as the groupoid associated to its corresponding labelled space (see [7] for groupoids associated to general labelled spaces). This can be done in terms of a shift map and the Renault-Deaconu construction [18,43].…”