Non Commutative Arens Algebras and their Derivations

Abstract

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau ,$ we consider the non commutative Arens algebra $L^{\omega }(M,\tau )=\bigcap _{p\ge 1}{L}^p(M,\tau )$ and the related algebras $L_2^{\omega }(M,\tau )=\bigcap _{p\ge 2}{L}^p(M,\tau )$ and $M+L_2^{\omega }(M,\tau )$ 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 $M+L_2^{\omega }(M,\tau )$ is inner and all derivations of the algebras $L^{\omega }(M,\tau )$ and $L_2^{\omega }(M,\tau )$ are spatial and implemented by elements of $M+L_2^{\omega }(M,\tau ).$

19 pages. Submitted to Journal of Functional analysis