Continuous derivations on algebras of locally measurable operators are inner

We prove that every derivation acting on the *‐algebra LS(M) of all locally measurable operators affiliated with a von Neumann algebra M is necessarily inner provided that it is continuous with respect to the local measure topology. In particular, every derivation on LS(M) is inner provided that M is a properly infinite von Neumann algebra. Furthermore, any derivation on an arbitrary von Neumann algebra M with values in a Banach M‐bimodule of locally measurable operators is inner.