Additive derivations on algebras of measurable operators

TL;DR

This paper introduces the central extension of a von Neumann algebra and investigates additive derivations, proving they are inner in certain classes of these algebras, thus advancing understanding of their algebraic structure.

Contribution

It defines the central extension of von Neumann algebras and proves that all additive derivations on these extensions are inner in specific cases.

Findings

mix(M) is a *-subalgebra of LS(M)

mix(M) equals LS(M) iff M has no type II summands

All additive derivations on mix(M) are inner for properly infinite M

Abstract

Given a von Neumann algebra $M$ we introduce so called central extension $mix(M)$ of $M$. We show that $mix(M)$ is a *-subalgebra in the algebra $LS(M)$ of all locally measurable operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and only if $M $ does not admit type II direct summands. We prove that if $M$ is a properly infinite von Neumann algebra then every additive derivation on the algebra $mix(M)$ is inner. This implies that on the algebra $LS(M)$, where $M$ is a type I$_\infty$ or a type III von Neumann algebra, all additive derivations are inner derivations.