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 (linear) 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.