Inverse semigroups associated with labelled spaces and their tight spectra

Boava, Giuliano; Castro, Gilles G. de; Mortari, Fernando de L.

doi:10.1007/s00233-016-9785-x

Cited by 18 publications

(45 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We say that a path µ is a prefix of a path ξ if ξ = µν for some path ν. 1 It is usual to use s instead of d and call it the source map, however we use Katsura's convention [7].…”

Section: The Boundary Path Space Of a Graphmentioning

confidence: 99%

Recovering the Boundary Path Space of a Topological Graph Using Pointless Topology

Castro¹

2020

J. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are T 1 and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

show abstract

“…We say that a path µ is a prefix of a path ξ if ξ = µν for some path ν. 1 It is usual to use s instead of d and call it the source map, however we use Katsura's convention [7].…”

Section: The Boundary Path Space Of a Graphmentioning

confidence: 99%

Recovering the Boundary Path Space of a Topological Graph Using Pointless Topology

Castro¹

2020

J. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is worth pointing out that in the case of a labelled space that is weakly left-resolving and normal, if a filter in a complete family is an ultrafilter, then all filters in the family coming before it are also ultrafilters [4,Proposition 5.7]. Theorem 4.13).…”

Section: Filters In E(s)mentioning

confidence: 99%

“…[4], Theorems 5.10 and 6.7). Let (E, L, B) be a weakly left-resolving, normal labelled space and S be its associated inverse semigroup.…”

mentioning

confidence: 96%

Groupoid Models for the C*-Algebra of Labelled Spaces

Boava

Castro

Mortari

2019

Bull Braz Math Soc, New Series

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In [5], the authors applied Exel's construction [7] to an inverse semigroup defined from a labelled space with multiplication inspired by the relations defining the C*-algebra of the labelled space. The tight spectrum was characterized in Theorem 6.7 of [5]. In the particular case of a labelled space defined from a graph as in [2], the authors found [5,Proposition 6.9] that the tight spectrum is homeomorphic to the boundary path space of the underlying graph (studied by Webster in [12]).…”

Section: Introductionmentioning

confidence: 99%

“…The tight spectrum was characterized in Theorem 6.7 of [5]. In the particular case of a labelled space defined from a graph as in [2], the authors found [5,Proposition 6.9] that the tight spectrum is homeomorphic to the boundary path space of the underlying graph (studied by Webster in [12]).…”

Section: Introductionmentioning

confidence: 99%

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

Boava

Castro

Mortari

2017

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Inverse semigroups associated with labelled spaces and their tight spectra

Cited by 18 publications

References 12 publications

Recovering the Boundary Path Space of a Topological Graph Using Pointless Topology

Recovering the Boundary Path Space of a Topological Graph Using Pointless Topology

Groupoid Models for the C*-Algebra of Labelled Spaces

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

Contact Info

Product

Resources

About

Inverse semigroups associated with labelled spaces and their tight spectra

Cited by 18 publications

References 12 publications

Recovering the Boundary Path Space of a Topological Graph Using Pointless Topology

Recovering the Boundary Path Space of a Topological Graph Using Pointless Topology

Groupoid Models for the C*-Algebra of Labelled Spaces

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

Contact Info

Product

Resources

About

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