2-Local derivations on algebras of locally measurable operators

The paper is devoted to 2-local derivations on the algebra $LS(M)$ of all locally measurable operators affiliated with a type I$_\infty$ von Neumann algebra $M.$ We prove that every 2-local derivation on $LS(M)$ is a derivation.Comment: arXiv admin note: substantial text overlap with arXiv:1110.155

Non Commutative Arens Algebras and their Derivations

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ we consider the non commutative Arens algebra $L^{\omega}(M, \tau)=\bigcap\limits_{p\geq1}L^{p}(M, \tau)$ and the related algebras $L^{\omega}_2(M, \tau)=\bigcap\limits_{p\geq2}L^{p}(M, \tau)$ and $M+L^{\omega}_2(M, \tau)$ which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra $M+L^{\omega}_2(M, \tau)$ is inner and all derivations of the algebras $L^{\omega}(M,\tau)$ and $L^{\omega}_2(M, \tau)$ are spatial and implemented by elements of $M+L^{\omega}_2(M, \tau).$Comment: 19 pages. Submitted to Journal of Functional analysi