Local and 2-Local derivations on noncommutative Arens algebras

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 $\tau \mathrm{.}$ 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 $L^\omega(M, \tau)$isalsoaspatialderivationandevery2-localderivationon$M$ is in fact an inner derivation.