Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers
arXiv:1908.09121
Abstract
The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type . 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--categories.
Invited contribution to the Proceedings of the 27th International Conference in Operator Theory (OT27), Timişoara, 2018