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 onMwith 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). © 2011 Elsevier Inc. All rights reserved