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

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.

Keywords:

*none*