Directed Graphs, von Neumann Algebras, and Index

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.

Free probability induced by electric resistance networks on energy Hilbert spaces

“…We then introduce a conditional expectation E of A G onto the subalgebra D G of diagonal elements. To get a representation of A G and an associated Hilbert space H G , we then use the Stinespring construction on E (e.g., see [5]). In this section, we introduce the concepts and definitions we use.…”

“…By considering groupoid elements as multiplication operators on certain Hilbert spaces, they become natural groupoid actions on Hilbert spaces. Such groupoid actions induce groupoid dynamical systems (e.g., [3,4,4,6], and [5]). …”

“…It is shown that the groupoid sum G 1 + G 2 of two graph groupoids G 1 and G 2 is groupoid-isomorphic to the graph groupoid G of the unioned graph G = G 1 ∪ G 2 in [5], where G k are the graph groupoids of G k , for k = 1, 2. Equivalently, the study of groupoid sums of graph groupoids is to investigate other graph groupoids determined by the unioned graphs.…”

“…In this section, we re-define electric resistance networks to apply our graph-groupoidal research (e.g., [1] through [5]). i.e., we construct graph groupoids induced by electric resistance networks and study certain operator algebras generated by electric resistance networks.…”

“…Independently, the first named author initiated an approach to graph groupoids based on free probability theory and representations (e.g., see [1] through [5]). …”