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In the first part of this paper, we give a new look at inclusions of vonNeumann algebras with finite-dimensional centers and finite Jones' index. Theminimal conditional expectation is characterized by means of a canonical stateon the relative commutant, that we call the spherical state; the minimal indexis neither additive nor multiplicative (it is submultiplicative), contrary tothe subfactor case. So we introduce a matrix dimension with the good functorialproperties: it is always additive and multiplicative. The minimal index turnsout to be the square of the norm of the matrix dimension, as was known in themulti-matrix inclusion case. In the second part, we show how our results arevalid in a purely 2-$C^*$-categorical context, in particular they can beformulated in the framework of Connes' bimodules over von Neumann algebras.

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