2016/05/18 by Jordá, Enrique, Peralta, Antonio M.
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1605.05656
Let Ω 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(Ω,A) is inner, then every weak-2-local derivation Δ:C(Ω,A)→ C(Ω,A) is a \rm(linear\rm) derivation. As a consequence we derive that, for every complex Hilbert space H, every weak-2-local derivation Δ: C(Ω,B(H)) → C(Ω,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(Ω,B) is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra M and every compact Hausdorff space Ω, every 2-local derivation on C(Ω,M) is a (linear) derivation.