Innerness of continuous derivations on algebras of measurable operators affiliated with finite von Neumann algebras

This paper is devoted to derivations on the algebra S(M) of all measurable operators affiliated with a finite von Neumann algebra M. We prove that if M is a finite von Neumann algebra with a faithful normal semi-finite trace τ, equipped with the locally measure topology t, then every t-continuous derivation D:S(M)→S(M) is inner. A similar result is valid for derivation on the algebra S(M,τ) of τ-measurable operators equipped with the measure topology tτ.