Derivations with values in noncommutative symmetric spaces

Abstract

Let $E=E(0,\infty )$ be a symmetric function space and $E(\mathcal{M},\tau )$ be the noncommutative symmetric space corresponding to $E(0,\infty )$ associated with a von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $\delta :\mathcal{A}\rightarrow E(\mathcal{M},\tau )$ 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.

Keywords

• derivation
• noncommutative symmetric space
• semifinite von Neumann algebra