Luogo e orario: aula 1B1, ore 17.00
Speaker: Luca Giorgetti
Titolo: Minimal index and dimension for 2-C*-categories with finite-dimensional centers
Abstract: The notion of index, in the sense we deal with in this talk, 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 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 matrix dimension, as it is the case for multi-matrices with respect to the Bratteli or inclusion matrix. We shall discuss the properties of this matrix dimension (multiplicativity and additivity, while the minimal index is neither multiplicative nor additive beyond the subfactor case). We show how the theory of minimal index can be formulated in the more general and purely algebraic context of 2-C*-categories.
Joint work with Roberto Longo (Roma Tor Vergata), preprint arxiv:1805.09234.