Innerness of Derivations on Subalgebras of Measurable Operators

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau ,$ let $L(M,\tau )$ be the algebra of all $\tau$-measurable operators affiliated with $M\mathrm{.}$ We prove that if $A$ is a locally convex reflexive complete metrizable solid $\ast$-subalgebra in $L(M,\tau ),$ which can be embedded into a locally bounded weak Fr'{e}chet $M$-bimodule, then any derivation on $A$ is inner.