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Graphs, groupoids, and Cuntz-Krieger algebras. (English) Zbl 0929.46055

Each locally finite directed graph \(G\), having only finitely many edges in and out of each vertex (but which may have infinitely many vertices), is associated with a locally compact groupoid \({\mathcal G}\), whose unit space is the space of one-sided infinite paths in \(G\). It is then shown that the groupoid \(C^*\)-algebra \(C^*({\mathcal G})\) of \({\mathcal G}\) is the universal \(C^*\)-algebra generated by (possibly infinite) families of partial isometries satisfying Cuntz-Krieger relations determined by \(G\). The structure theory for groupoid \(C^*\)-algebras, as developed by Renault, is used to analyze the ideal structure of Cuntz-Krieger algebras, leading, in turn, to extensions of the results of Cuntz and Cuntz-Krieger to the case of infinite, locally finite \(\{0,1\}\)-matrices. By choosing a distinguished vertex \(*\), one obtains, as a reduction of \({\mathcal G}\) to the space of paths emanating from \(*\), another locally compact groupoid \(C^*({\mathcal G}(*))\). Under certain conditions, the \(C^*\)-algebras \(C^*({\mathcal G})\) and \(C^*({\mathcal G}(*))\) are Morita equivalent. Moreover, the algebra \(C^*({\mathcal G}(*))\) contains the \(C^*\)-algebra used by Doplicher and Roberts in their duality theory for compact groups.

MSC:

46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
05C20 Directed graphs (digraphs), tournaments
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