Local derivations on measurable operators and commutativity

TL;DR

This paper characterizes abelian von Neumann algebras through properties of derivations on measurable operators, revealing a unique link between algebraic structure and derivation behavior.

Contribution

It establishes a new characterization of abelian von Neumann algebras via the behavior of squared derivations on measurable operators.

Findings

Square of every derivation on $S(M)$ is a local derivation if and only if $M$ is abelian.

The characterization does not extend to general associative unital algebras.

Provides insight into the structure of derivations in operator algebras.

Abstract

We prove that a von Neumann algebra $M$ is abelian if and only if the square of every derivation on the algebra $S(M)$ of measurable operators, affiliated with $M$, is a local derivation. We also show that for general associative unital algebras this is not true.