Additive derivations on generalized Arens algebras

TL;DR

This paper characterizes all additive derivations on generalized Arens algebras associated with von Neumann algebras, showing that for type II algebras, all such derivations are inner, thus deepening understanding of their algebraic structure.

Contribution

It provides a complete description of additive derivations on generalized Arens algebras, proving that they are inner for type II von Neumann algebras.

Findings

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

For type II von Neumann algebras, all additive derivations are inner.

The paper advances the understanding of derivations on generalized Arens algebras.

Abstract

Given a von Neumann algebra $M$ with a faithful normal finite trace $\tau$ denote by $L^\Lambda(M, \tau)$ the generalized Arens algebra with respect to $M.$ We give a complete description of all additive derivations on the algebra $L^\Lambda(M, \tau).$ In particular each additive derivation on the algebra $L^{\Lambda}(M, \tau),$ where $M$ is a type II von Neumann algebra, is inner.