Graph inverse semigroups, groupoids and their
| Venue: | C ∗ -algebras, J. Operator Theory |
| Citations: | 50 - 2 self |
BibTeX
@ARTICLE{Paterson_graphinverse,
author = {Alan L. T. Paterson},
title = {Graph inverse semigroups, groupoids and their},
journal = {C ∗ -algebras, J. Operator Theory},
year = {}
}
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Abstract
Abstract. There is now a substantial literature on graph C ∗-algebras. Under a locally finite condition on a countable, directed graph, Kumjian, Pask, Raeburn, Renault showed that the C ∗-algebra of the graph can be realized as the C ∗-algebra of the path groupoid, i.e. the groupoid determined by the infinite paths in the graph. In the present paper, we remove the local finiteness requirement. The path groupoid in the general context is obtained through the universal groupoid of a certain inverse semigroup associated with the graph. This inverse semigroup is called the graph inverse semigroup, and graph representations turn out to be just representations of this inverse semigroup. A certain reduction of the universal groupoid gives the path groupoid of the graph, and its C ∗-algebra is isomorphic to the C ∗-algebra of the graph. The unit space of the path groupoid contains the infinite paths of the graph, but also contains some finite paths. We show that, as in the locally finite case, the path groupoid is always amenable, and we give a groupoid proof of a recent theorem of W. Szymanski, characterizing when a graph C ∗-algebra is simple. 1.
Keyphrases
path groupoid graph inverse semigroups infinite path inverse semigroup graph algebra universal groupoid certain reduction certain inverse semigroup general context substantial literature finite path recent theorem groupoid proof finite case graph inverse semigroup unit space present paper finite condition local finiteness requirement graph representation
