The Index Semigroup and K-Groups of Graph-Groupoid C~*-Algebras

摘要

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 pie, 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.