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?1Lp(M,τ) and the related algebras and 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 is inner and all derivations of the algebras Lω(M,τ) and are spatial and implemented by elements of . In particular we obtain that if the trace τ is finite then any derivation on the non-commutative Arens algebra Lω(M,τ) is inner.