Commutator estimates in $W^{\ast }$-algebras

Abstract

Let $\mathcal{M}$ be a $W^{\ast }$-algebra and let $LS(\mathcal{M})$ be the algebra of all locally measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in LS(\mathcal{M})$ there exists a self-adjoint element $c_{{}_0}$ from the center of $LS(\mathcal{M})$, such that for any $\epsilon >0$ there exists a unitary element $u_{\epsilon }$ from $\mathcal{M}$, satisfying $|[a,u_{\epsilon }]|\ge (1-\epsilon )|a-c_{{}_0}|$. A corollary of this result is that for any derivation $\delta$ on $\mathcal{M}$ with the range in a (not necessarily norm-closed) ideal $I\subseteq \mathcal{M}$, the derivation $\delta$ is inner, that is $\delta (\cdot )={\delta }_a(\cdot )=[a,\cdot ]$, and $a\in I$. Similar results are also obtained for inner derivations on $LS(\mathcal{M})$.

30 pages