Local and 2-Local derivations on noncommutative Arens algebras

TL;DR

This paper investigates local and 2-local derivations on noncommutative Arens algebras, proving that under certain conditions these derivations are spatial or inner, thus characterizing their structure.

Contribution

It establishes that all 2-local derivations on these algebras are spatial and, in finite von Neumann cases, are inner derivations, extending understanding of derivation structures.

Findings

Every 2-local derivation on $L^(M, au)$ is spatial.

On finite von Neumann algebras, local derivations are spatial.

2-local derivations on finite von Neumann algebras are inner.

Abstract

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra $L^\omega(M, \tau)$ associated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $\tau.$ We prove that every 2-local derivation on $L^\omega(M, \tau)$ is a spatial derivation, and if $M$ is a finite von Neumann algebra, then each local derivation on $L^\omega(M, \tau)$ is also a spatial derivation and every 2-local derivation on $M$ is in fact an inner derivation.