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The Index Semigroup and \(K\)-Groups of Graph-Groupoid \(C^{*}\)-Algebras

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Abstract

In this paper, we consider the relation between index theory and \(K\)-theory induced by directed graphs. In particular, we study index-morphism on finite trees, and classify the set of finite trees in terms of our index-morphism. Such a morphism generate certain semigroup, called the index semigroup. From the index semigroup, we find a ple, interesting connection between semigroup-elements and \(K\)-group computations of groupoid \(C^{*}\)-algebras generated by graphs. In conclusion, we show that the pure combinatorial data of graphs completely characterize and classify the elements of the index semigroup (or equivalently, graph-index on finite trees), Watatani’s Jones index on groupoid \(C^{*}\)-algebras generated by finite trees, and \(K\)-group computations of certain \(C^{*}\)-algebras.

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Correspondence to Ilwoo Cho.

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Communicated by Palle Jorgensen.

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Cho, I. The Index Semigroup and \(K\)-Groups of Graph-Groupoid \(C^{*}\)-Algebras. Complex Anal. Oper. Theory 8, 57–109 (2014). https://doi.org/10.1007/s11785-012-0271-5

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