Minimal Index and Dimension for 2C*Categories with FiniteDimensional Centers
Abstract
In the first part of this paper, we give a newlook at inclusions of von Neumann algebras with finitedimensional 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 multimatrix 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 September 2019
 DOI:
 10.1007/s002200183266x
 arXiv:
 arXiv:1805.09234
 Bibcode:
 2019CMaPh.370..719G
 Keywords:

 Mathematics  Operator Algebras;
 Mathematical Physics;
 Mathematics  Category Theory;
 46L37;
 18D10;
 46L10
 EPrint:
 38 pages