Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras

TL;DR

This paper proves that all continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras are inner, extending the understanding of derivation structure in operator algebras.

Contribution

It establishes that every measure-topology continuous derivation on these algebras is inner, a significant generalization in the theory of operator algebras.

Findings

All $t$-continuous derivations are inner.

Results apply to algebras with faithful normal semi-finite trace.

Extends to derivations on $ au$-measurable operators.

Abstract

This paper is devoted to derivations on the algebra $S(M)$ of all measurable operators affiliated with a finite von Neumann algebra $M.$ We prove that if $M$ is a finite von Neumann algebra with a faithful normal semi-finite trace $\tau$, equipped with the locally measure topology $t,$ then every $t$-continuous derivation $D:S(M)\rightarrow S(M)$ is inner. A similar result is valid for derivation on the algebra $S(M,\tau)$ of $\tau$-measurable operators equipped with the measure topology $t_{\tau}$.