Computing the Jones index of quadratic permutation endomorphisms of O_2

TL;DR

This paper calculates the Jones index for certain type III factors derived from endomorphisms of the Cuntz algebra O_2 linked to rank-two permutation matrices, advancing understanding of subfactor theory.

Contribution

It provides explicit index computations for quadratic permutation endomorphisms of O_2, a novel contribution to the study of subfactors and operator algebras.

Findings

Computed the Jones index for specific endomorphisms of O_2

Identified the type III_{1/2} factors arising from these endomorphisms

Enhanced understanding of the structure of subfactors related to permutation matrices

Abstract

We compute the index of the inclusions of type $III_{1/2}$ factors arising from endomorphisms of the Cuntz algebra ${\mathcal O}_2$ associated to the rank-two permutation matrices.