An Inclusion of Type III Factors with Index 4 Arising from an Automorphism

Abstract

We consider an inclusion of type $\mathrm{III}$ factors with index 4 arising from an automorphism. It has the extended Coxeter–Dynkin diagram $\tilde{A}$ or $A_{\infty ,\infty }$ as the principal graph. We show the equivalence between classification of automorphisms and that of the corresponding subfactors, and compute conjugacy invariants for the subfactors. As an application, we show the existence of a pair of type ${\mathrm{III}}_1$ factors which does not split into a type ${\mathrm{II}}_1$ inclusion.