Approximate innerness and central triviality of endomorphisms

TL;DR

This paper introduces and characterizes approximate innerness and central triviality for endomorphisms on separable von Neumann factors, generalizing known results from automorphisms using Connes-Takesaki modules and modular endomorphisms.

Contribution

It extends the concepts of approximate innerness and central triviality to endomorphisms, providing a characterization for hyperfinite factors that generalizes previous automorphism results.

Findings

Characterization of approximate innerness and central triviality for hyperfinite factors

Use of Connes-Takesaki modules and modular endomorphisms in the characterization

Generalization of automorphism results to endomorphisms

Abstract

We introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by Kawahigashi-Sutherland-Takesaki in automorphism case.