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 t τ .