Commutator estimates in $W^*$-factors

Abstract:

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 $S(\mathcal {M})$.