Inner Derivations and Weak-2-Local Derivations on the $$\hbox {C}^*$$ C ∗ -Algebra $$\varvec{C_0(L,A)}$$ C 0 ( L , A )

Abstract

Let L be a locally compact Hausdorff space. Suppose A is a $$\hbox {C}^*$$ -algebra with the property that every weak-2-local derivation on A is a (linear) derivation. We prove that every weak-2-local derivation on $$C_0(L,A)$$ is a (linear) derivation. Among the consequences we establish that if B is an atomic von Neumann algebra or a compact $$\hbox {C}^*$$ -algebra, then every weak-2-local derivation on $$C_0(L,B)$$ is a linear derivation. We further show that, for a general von Neumann algebra M, every 2-local derivation on $$C_0(L,M)$$ is a linear derivation. We also prove several results representing derivations on $$C_0(L,B(H))$$ and on $$C_0(L,K(H))$$ as inner derivations determined by multipliers.