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We prove that every derivation acting on the   *  ‐algebra    L  S  (  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    L  S  (  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.

Continuous derivations on algebras of locally measurable operators are inner