Innerness of derivations into noncommutative symmetric spaces is determined commutatively

TL;DR

This paper characterizes when derivations into noncommutative symmetric spaces are inner, showing it depends on the underlying symmetric function space having the Levi property.

Contribution

It identifies the class of symmetric function spaces where all derivations into associated noncommutative spaces are inner, based on the Levi property.

Findings

Derivations are inner iff the symmetric function space has the Levi property.

The result applies to all semifinite von Neumann algebras.

Provides a complete characterization of inner derivations in this setting.

Abstract

Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},\tau)$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $\delta:\mathcal{A}\to E(\mathcal{M},\tau)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.