Approximate Unitary Equivalence of Finite Index Endomorphisms of the AFD Factors

TL;DR

This paper characterizes when two finite index endomorphisms of AFD factors are approximately unitarily equivalent, using canonical extensions and recent developments in the Rohlin property, generalizing previous results.

Contribution

It provides a new characterization of approximate unitary equivalence for finite index endomorphisms on AFD factors, independent of factor types.

Findings

Uses canonical extension of endomorphisms for characterization

Generalizes previous results on approximate innerness

Employs Rohlin property of flows in the proof

Abstract

We characterize the condition for two finite index endomorphisms on an AFD factor to be approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.