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.