  Let $\mathcal{M}$ be a $W^*$-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 $\varepsilon > 0$, there
exists a unitary element $u_\varepsilon$ from $\mathcal{M}$, satisfying
$|[a,u_\varepsilon]| \geq (1-\varepsilon)|a-\lambda_0\mathbf{1}|$. 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 $S(\mathcal{M})$.
