×

A theory of dimension. (English) Zbl 0874.18005

The purpose of this paper is to develop a theory of dimension which is related to the Jones index and based on the notion of conjugation. Namely, let \({\mathcal T}\) be a strict monoidal C*-category in the sense from S. Doplicher and J. E. Roberts [“A new duality theory for compact groups”, Invent. Math. 98, No. 1, 157-218 (1989; Zbl 0691.22002)], with objects denoted by \(\rho,\sigma,\tau\), etc., which has a complex Banach \((\rho, \sigma)\) as the space of arrows between \(\rho\) and \(\sigma\), the composition of arrows denoted by “\(\circ\)” is bilinear, the adjoint \(*\) is an involutive contravariant functor acting as the identity on objects and has an associative bilinear bifunctor \(\otimes: {\mathcal T} \times {\mathcal T} \to {\mathcal T}\), with a unit \(i\) and commuting with \(*\).
The authors consider also a full subcategory \({\mathcal T}_f\) of \({\mathcal T}\) as follows: an object \(\rho\) of \({\mathcal T}\) lies in \({\mathcal T}_f\) if there is an object \(\overline\rho\) of \({\mathcal T}\), the conjugate of \(\rho\), and \(R\in (t,\overline \rho \otimes \rho)\) and \(\overline R \in(c, \rho \otimes \overline \rho)\) such that \(\overline R^* \otimes 1_\rho \circ 1_\rho \otimes R=1_\rho\), \(R^* \otimes 1_{\overline \rho} \circ 1_{\overline \rho} \otimes \overline R=1_{\overline \rho}\). The essential step towards introducing a dimension for the objects of \({\mathcal T}_f\) rather than their left inverses is to choose a special equivalence class of solutions of the previous conjugate equations. So, the notion of the standard left inverse is obtained and the dimension \(d(\rho)\) for an object \(\rho\) is defined as the dimension of its standard left inverse. Hence \(d(\rho)= d(\overline \rho)\) so that \(d(\rho)\) depends only on the unitary equivalence class of \(\rho\) and is additive on direct sums. Also, an elementary proof of the multiplicativity of the dimension on tensor products is given.
Applications to a class of endomorphisms of factors and to the theory of subfactors are given. Here an important role is played by a notion of amenability inspired by Popa’s notion of a strongly amenable inclusion of factors.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
46L37 Subfactors and their classification
81T05 Axiomatic quantum field theory; operator algebras

Citations:

Zbl 0691.22002
PDFBibTeX XMLCite
Full Text: DOI arXiv