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

Abstract

Let $\mathcal{M}$ be a $W^{\ast }$-factor and let $S\left(\mathcal{M}\right)$ be the space of all measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in S(\mathcal{M})$ there exists a scalar ${\lambda }_0\in \mathbb{R}$, such that for all $\epsilon >0$, there exists a unitary element $u_{\epsilon }$ from $\mathcal{M}$, satisfying $|[a,u_{\epsilon }]|\ge (1-\epsilon )|a-{\lambda }_{0}1|$. A corollary of this result is that for any derivation $\delta$ on $\mathcal{M}$ with the range in an 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 $S(\mathcal{M})$.

21 pages