# Minimal index and dimension for 2-C*-categories with finite-dimensional centers

Luca Giorgetti, Roberto Longo
May 23, 2018
In the first part of this paper, we give a new look at inclusions of von Neumann algebras with finite-dimensional centers and finite Jones' index. The minimal conditional expectation is characterized by means of a canonical state on the relative commutant, that we call the spherical state; the minimal index is neither additive nor multiplicative (it is submultiplicative), contrary to the subfactor case. So we introduce a matrix dimension with the good functorial properties: it is always additive and multiplicative. The minimal index turns out to be the square of the norm of the matrix dimension, as was known in the multi-matrix inclusion case. In the second part, we show how our results are valid in a purely 2-C*-categorical context, in particular they can be formulated in the framework of Connes' bimodules over von Neumann algebras.
@article{Giorgetti:2018nji, author = "Giorgetti, Luca and Longo, Roberto", title = "{Minimal Index and Dimension for 2-$C^*$-Categories with Finite-Dimensional Centers}", journal = "Commun. Math. Phys.", volume = "370", year = "2019", number = "2", pages = "719-757", doi = "10.1007/s00220-018-3266-x", eprint = "1805.09234", archivePrefix = "arXiv", primaryClass = "math.OA", SLACcitation = "%%CITATION = ARXIV:1805.09234;%%" }