Innerness of Derivations on Subalgebras of Measurable Operators

TL;DR

This paper proves that derivations on certain subalgebras of measurable operators affiliated with a von Neumann algebra are always inner, under specific topological and algebraic conditions.

Contribution

It establishes that derivations on locally convex reflexive subalgebras of measurable operators are inner if they can be embedded into a locally bounded weak Fréchet bimodule.

Findings

Derivations on the specified subalgebras are inner.

The result applies to subalgebras with particular topological properties.

The proof involves embedding into weak Fréchet bimodules.

Abstract

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ let $L(M, \tau)$ be the algebra of all $\tau$-measurable operators affiliated with $M.$ We prove that if $A$ is a locally convex reflexive complete metrizable solid $\ast$-subalgebra in $L(M, \tau),$ which can be embedded into a locally bounded weak Fr\'{e}chet $M$-bimodule, then any derivation on $A$ is inner.