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) = boolean AND(p >= 1) L-p(M, tau) and the related algebras L-2(omega)(M, tau) = boolean AND(p >= 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). In particular we obtain that if the trace tau is finite then any derivation on the non-commutative Arens algebra L-omega(M, tau) is inner. (c) 2007 Elsevier Inc. All rights reserved.