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

TL;DR

This paper studies inclusions of von Neumann algebras with finite-dimensional centers, introducing a matrix dimension with good properties and relating it to the minimal index, extending results to 2-$C^*$-categories and bimodules.

Contribution

It introduces a functorial matrix dimension for finite-dimensional center inclusions and relates it to the minimal index, extending the theory to 2-$C^*$-categories.

Findings

The minimal index is the square of the matrix dimension's norm.

The matrix dimension is additive and multiplicative.

Results are applicable in the context of Connes' bimodules.

Abstract

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.