Commutator estimates in $W^*$-algebras

TL;DR

This paper establishes estimates for commutators in $W^*$-algebras, showing that self-adjoint elements can be approximated by central elements with controlled commutator behavior, and proves that derivations into certain ideals are inner.

Contribution

It introduces new commutator estimates in $W^*$-algebras and demonstrates that derivations into ideals are necessarily inner, extending understanding of algebraic structure and derivation properties.

Findings

Existence of a central element close to any self-adjoint element with controlled commutator bounds

Derivations into ideals are inner, represented by commutators with an element in the ideal

Similar results hold for inner derivations on $LS(rak{M})$

Abstract

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 $LS(\mathcal{M})$.