GRAPH INVERSE SEMIGROUPS, GROUPOIDS AND THEIR C -ALGEBRAS

By A. Paterson. 2003

136 citations recorded by the source corpus.

We develop a theory of graph C -algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges. We show that the path groupoid is amenable, and give a groupoid proof of a recent theorem of Szymanski characterizing when a graph C - algebra is simple.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/97dfbe78cd0764c0b9851a0d9a675158806f9f8b

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

The path space of a directed graph (2011, methodology): “2] and has since appeared in more general settings, for example [7,8,11].”
ON AN ORDER-BASED CONSTRUCTION OF A TOPOLOGICAL GROUPOID FROM AN INVERSE SEMIGROUP (2008, result): “results of [ 9 ] concerning the structure of the open invariant subsets of G ( g ) (0) (see [ 14 ] for a recent extension to a non-locally-ﬁnite situation).”
The ideal structure of the $C\sp *$-algebras of infinite graphs (2001, methodology): “First, they provide good test problems in the general theories of groupoid algebras [16, 15, 18 ], Cuntz-Pimsner algebras [19, 21, 13, 9, 10], and partial crossed products [6, 8]. Second, the simple graph algebras provide a rich family of accessible models for purely infinite simple C ∗-algebras.”
KMS states on the C∗-algebras of finite graphs (2012, methodology): “To avoid having to adjust the standard groupoid model of [10] or [16], we assume that E has no sources.”
The $C^*$-Algebras of Arbitrary Graphs (2000, background): “It was not until 2000 that Fowler, Laca, and Raeburn were finally able to extend the definition of graph algebras to arbitrary directed graphs [8].”
Graded Steinberg algebras and their representations (2017, methodology): “We recall from [Clark and Sims 2015, Example 2.1] the construction of the groupoid G E from an arbitrary graph E , which was introduced in [Kumjian et al. 1997] for row-ﬁnite graphs and generalised to arbitrary graphs in [Paterson 2002].”
Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach (2004, unknown): “O A ] 2 2 M ar”
Inverse semigroups and combinatorial C*-algebras (2007, unknown): “*Partially supported by CNPq.”