Description of Derivations on Measurable Operator Algebras of Type I

TL;DR

This paper characterizes all derivations on the algebra of measurable operators affiliated with type I von Neumann algebras, showing that for type I$_\infty$, all derivations are inner, thus providing a complete structural understanding.

Contribution

It provides a complete description of derivations on $L(M, au)$ for type I von Neumann algebras, especially proving innerness for type I$_\infty$ cases.

Findings

All derivations on $L(M, au)$ are characterized.

Derivations on type I$_\infty$ are inner.

Complete description of derivations on measurable operator algebras.

Abstract

Given a type I 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 give a complete description of all derivations on the algebra $L(M, \tau).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $L(M, \tau)$ is inner.