Non-commutative Arens algebras and their derivations

Given a von Neumann algebra M with a faithful normal semi-finite trace τ , we consider the non-commutative Arens algebra Lω(M, τ ) = p 1 Lp(M, τ ) and the related algebras Lω2 (M, τ ) = p 2 Lp(M, τ ) andM+Lω2 (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 + Lω2 (M, τ ) is inner and all derivations of the algebras Lω(M, τ ) and Lω2 (M, τ ) are spatial and implemented by elements of M + Lω2 (M, τ ). In particular we obtain that if the trace τ is finite then any derivation on the noncommutative Arens algebra Lω(M, τ ) is inner. © 2007 Elsevier Inc. All rights reserved.