Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

TL;DR

This paper characterizes derivations on various algebras of measurable operators affiliated with type I von Neumann algebras, showing they are inner or spatial under certain conditions.

Contribution

It provides a complete description of derivations on algebras of measurable operators for type I von Neumann algebras, including conditions for inner and spatial derivations.

Findings

Derivations on LS(M), S(M), and S(M, τ) are inner for type I_∞ von Neumann algebras.

Derivations on S_0(M, τ) are spatial and implemented by elements from S(M, τ).

The results extend the understanding of algebraic structures of measurable operators in operator algebras.

Abstract

Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $\tau$ on $M$ let $S(M, \tau)$ (resp. $S_0(M, \tau)$) be the algebra of all $\tau$-measurable (resp. $\tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,\tau)$) is inner, and each derivation on $S_0(M, \tau)$ is spatial and implemented by an element from $S(M, \tau).$