Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers

Luca Giorgetti
August 24, 2019
The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type II_1. In the absence of a trace, one can still define the index of a conditional expectation associated to a subfactor and look for expectations that minimize the index. This value is called the minimal index of the subfactor. We report on our analysis, contained in [GL19], of the minimal index for inclusions of arbitrary von Neumann algebras (not necessarily finite, nor factorial) with finite-dimensional centers. Our results generalize some aspects of the Jones index for multi-matrix inclusions (finite direct sums of matrix algebras), e.g., the minimal index always equals the squared norm of a matrix, that we call \emph{matrix dimension}, as it is the case for multi-matrices with respect to the Bratteli inclusion matrix. We also mention how the theory of minimal index can be formulated in the purely algebraic context of rigid 2-C^*-categories.
open access link
@inproceedings{Giorgetti:2019mrh, author = "Giorgetti, Luca", title = "{Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers}", booktitle = "{27th International Conference in Operator Theory (OT27) Timisoara, Romanida, July 2-6, 2018}", year = "2019", eprint = "1908.09121", archivePrefix = "arXiv", primaryClass = "math.OA", reportNumber = "Roma01.Math", SLACcitation = "%%CITATION = ARXIV:1908.09121;%%" }

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