A Theory of Dimension

TL;DR

This paper develops a theoretical framework for dimension related to the Jones index, introducing conjugation-based concepts, and explores applications to endomorphisms, subfactors, and amenability in operator algebras.

Contribution

It presents a new theory of dimension based on conjugation, with elementary proofs of key properties and applications to subfactor theory and amenability.

Findings

Elementary proof of additivity and multiplicativity of dimension

Introduction of an associated trace for the dimension theory

Applications to endomorphisms of factors and subfactor theory

Abstract

In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to the theory of subfactors. An important role is played by a notion of amenability inspired by the work of Popa.