Stability of derivations under weak-2-local continuous perturbations

Abstract

Let $\Omega$ be a compact Hausdorff space and let $A$ be a C$^*$-algebra. We prove that if every weak-2-local derivation on $A$ is a linear derivation and every derivation on $C(\Omega,A)$ is inner, then every weak-2-local derivation $\Delta:C(\Omega,A)\to C(\Omega,A)$ is a {\rm(}linear{\rm)} derivation. As a consequence we derive that, for every complex Hilbert space $H$, every weak-2-local derivation $\Delta : C(\Omega,B(H)) \to C(\Omega,B(H))$ is a (linear) derivation. We actually show that the same conclusion remains true when $B(H)$ is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if $B$ denotes a compact C$^*$-algebra (in particular, when $B=K(H)$), then every weak-2-local derivation on $C(\Omega,B)$ is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra $M$ and every compact Hausdorff space $\Omega$, every 2-local derivation on $C(\Omega,M)$ is a (linear) derivation.