The Index Semigroup Induced by Finite Trees

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The study focuses on the relationship between graph-index and Watatani's extended Jones index, particularly in the context of finite trees. It demonstrates the equivalence of these indices for graphs and von Neumann algebras generated by graph groupoids. The classification of graph-groupoid von Neumann algebras based on these indices is provided, alongside the introduction of "index semigroup" as a new algebraic structure. Additionally, the work explores the connection between the index semigroup and K-groups of finite-tree-groupoid von Neumann algebras, highlighting fundamental properties and applications.