Continuous derivations on algebras of locally measurable operators are inner (Q2874664)

\textit{J. R. Ringrose} [J. Lond. Math. Soc., II. Ser. 5, 432--438 (1972; Zbl 0245.46084)] showed that any derivation acting on a \(C^*\)-algebra \(\mathcal{M}\) with values in a Banach \(\mathcal{M}\)-bimodule is automatically norm-continuous. In the paper under review, the authors generalize this fact by showing that for every derivation \(\delta\) acting on the *-algebra \(LS(\mathcal{M})\) of all locally measurable operators affiliated with a von Neumann algebra \(\mathcal{M}\), which is continuous with respect to the local measure topology \(\tau(\mathcal{M})\), there exists \(a \in LS(\mathcal{M})\) such that \(\delta(x)= ax-xa\) for all \(x\in LS(\mathcal{M})\), that is, the derivation \(\delta\) is inner. In particular, every derivation on \(LS(\mathcal{M})\) is inner provided that \(\mathcal{M}\) is a properly infinite von Neumann algebra, see [\textit{A. F. Ber} et al., Integral Equations Oper. Theory 75, No. 4, 527--557 (2013; Zbl 1304.46063)]. Furthermore, the authors prove that every derivation \(\delta\) on an arbitrary von Neumann algebra \(\mathcal{M}\) with values in a Banach \(\mathcal{M}\)-bimodule \(\mathcal{E}\) of locally measurable operators is inner. In particular, \(\delta\) is a continuous derivation from \((\mathcal{M}, \|\cdot\|_\mathcal{M})\) into \((\mathcal{E}, \|\cdot\|_\mathcal{E})\).