Commutator estimates in W⁎-algebras

Let M be a W⁎-algebra and let LS(M) be the algebra of all locally measurable operators affiliated with M. It is shown that for any self-adjoint element a∈LS(M) there exists a self-adjoint element c0 from the center of LS(M), such that for any ε>0 there exists a unitary element uε from M, satisfying |[a,uε]|⩾(1−ε)|a−c0|. A corollary of this result is that for any derivation δ on M with the range in a (not necessarily norm-closed) ideal I⊆M, the derivation δ is inner, that is δ(⋅)=δa(⋅)=[a,⋅], and a∈I. Similar results are also obtained for inner derivations on LS(M).