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We prove that any derivation of the * -algebra LS(M) of all locally measurable operators affiliated with a properly infinite von Neumann algebra M is continuous with respect to the local measure topology t(M). Building an extension of a derivation δ : M −→ LS(M) up to a derivation from LS(M) into LS(M), it is further established that any derivation from M into LS(M) is t(M)-continuous.

Abstract. It is established that every derivation continuous with respect to the local measure topology acting on the * -algebra LS(M) of all locally measurable operators affiliated with a von Neumann algebra M is necessary inner. If M is a properly infinite von Neumann algebra, then every derivation on LS(M) is inner. In addition, it is proved that any derivation on M with values in Banach M-bimodule of locally measurable operators is inner.

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Abstract. Let M be a W * -factor and let S (M) be the space of all measurable operators affiliated with M. It is shown that for any self-adjoint element a ∈ S(M) there exists a scalar λ 0 ∈ R, such that for all ε > 0, there exists a unitary element u ε from M, satisfying |[a, u ε ]| ≥ (1 − ε)|a − λ 0 1|. A corollary of this result is that for any derivation δ on M with the range in an ideal I ⊆ M, the derivation δ is inner, that is δ(·) = δ a (·) = [a, ·], and a ∈ I. Similar results are also obtained for inner derivations on S(M).