Innerness of continuous derivations on algebras of locally measurable operators

TL;DR

The paper proves that all continuous derivations on algebras of locally measurable operators affiliated with von Neumann algebras are inner, extending to all derivations on such algebras under certain conditions.

Contribution

It establishes that every derivation continuous in the local measure topology is inner, and all derivations on $LS(rak{M})$ are inner for properly infinite von Neumann algebras.

Findings

Continuous derivations are necessarily inner.

All derivations on $LS(rak{M})$ are inner for properly infinite $rak{M}$.

Derivations with values in Banach bimodules are inner.

Abstract

It is established that every derivation continuous with respect to the local measure topology acting on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ is necessary inner. If $\mathcal{M}$ is a properly infinite von Neumann algebra, then every derivation on $LS(\mathcal{M})$ is inner. In addition, it is proved that any derivation on $\mathcal{M}$ with values in Banach $\mathcal{M}$-bimodule of locally measurable operators is inner.