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

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.

open access link
doi:10.1007/s00220-018-3266-x

@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;%%"
}

Keywords:

*none*