Innerness of Derivations on Subalgebras of Measurable Operators

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

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