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 $M.$ 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.