The index semigroup and \(K\)-groups of graph-groupoid \(C^{*}\)-algebras. (English) Zbl 1297.05096
Summary: 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.
MSC:
| 05C20 | Directed graphs (digraphs), tournaments |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |
| 46N50 | Applications of functional analysis in quantum physics |
| 47N99 | Miscellaneous applications of operator theory |
| 68Q12 | Quantum algorithms and complexity in the theory of computing |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
Keywords:
finite directed graphs; graph groupoids; graph-von Neumann algebras; graph inclusions; graph index; Jones index; quasi-bases; Watatani’s extended Jones index; index morphism; index semigroupsReferences:
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