Abstract
In this paper, we assign index numbers to finite directed graphs. Motivated by the indices of Jones and Watatani (from operator algebra theory), we introduce and compute a new graph-theoretical index, and consider the connection with Watatani’s extended Jones index. Starting with an inclusion of finite directed graphs, we show that there is a natural subgroupoid inclusion, and then a tower of von Neumann algebras. In particular, each step in the tower having the same index number, under certain normalization.
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The second named author is supported by the U. S. National Science Foundation.
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Cho, I., Jorgensen, P.E.T. Directed Graphs, von Neumann Algebras, and Index. Algebr Represent Theor 15, 53–108 (2012). https://doi.org/10.1007/s10468-010-9233-7
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DOI: https://doi.org/10.1007/s10468-010-9233-7
Keywords
- Finite directed graphs
- Graph groupoids
- Graph von Neumann algebras
- Graph inclusions
- Graph-index
- Jones index
- Quasi-bases
- Watatani’s extended Jones index