# 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.

## Key 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.