Non Commutative Arens Algebras and their Derivations

Abstract

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ we consider the non commutative Arens algebra $L^ω(M, τ)=\bigcap\limits_{p\geq1}L^{p}(M, τ)$ and the related algebras $L^ω_2(M, τ)=\bigcap\limits_{p\geq2}L^{p}(M, τ)$ and $M+L^ω_2(M, τ)$ 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 $M+L^ω_2(M, τ)$ is inner and all derivations of the algebras $L^ω(M,τ)$ and $L^ω_2(M, τ)$ are spatial and implemented by elements of $M+L^ω_2(M, τ).$