Minimal Index and Dimension for 2-C*-Categories with Finite-Dimensional Centers
In the first part of this paper, we give a newlook 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 amatrix dimension with the good functorial properties: it is always additive and multiplicative. Theminimal 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.
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We thank the Isaac Newton Institute (INI) for Mathematical Sciences in Cambridge for hospitality during the program “Operator algebras: subfactors and their applications” (OAS), supported by EPSRC Grant Nos. EP/K032208/1 and EP/R014604/1. L.G. thanks also the Department of Mathematics of the Ohio University and of the Georg-August-Universität Göttingen for financial support, the latter during the workshop LQP41 “Foundations and constructive aspects of QFT” where part of our results were presented. We acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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