Commutator estimates in $W^*$-factors

TL;DR

This paper proves that in $W^*$-factors, self-adjoint measurable operators can be approximated by unitaries to analyze derivations, showing they are inner and related to the ideal structure.

Contribution

It establishes commutator estimates for self-adjoint operators in $W^*$-factors and demonstrates that derivations with range in an ideal are necessarily inner.

Findings

Existence of unitaries approximating self-adjoint operators

Derivations with range in an ideal are inner

Results extend to inner derivations on $S(\\mathcal{M})$

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 $S(\mathcal{M})$.