  Let $\mathcal{M}$ be a $W^*$-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 $|[a,u_\epsilon]| \geq (1-\epsilon)|a-c_{_{0}}|$. 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 $LS(\mathcal{M})$.
