Inner derivations and weak-2-local derivations on the C$^*$-algebra $C_0(L,A)$

TL;DR

This paper investigates the structure of weak-2-local derivations on C*-algebras of the form C_0(L,A), showing they are linear derivations under certain conditions, extending known results to various classes of operator algebras.

Contribution

It proves that weak-2-local derivations on C_0(L,A) are linear derivations when A has this property, generalizing previous results to broader classes of C*-algebras and von Neumann algebras.

Findings

Weak-2-local derivations on C_0(L,A) are linear derivations for certain A.

All weak-2-local derivations on C_0(L,B) are linear when B is atomic von Neumann or compact C*-algebra.

2-local derivations on C_0(L,M) are linear for general von Neumann algebra M.

Abstract

Let $L$ be a locally compact Hausdorff space. Suppose $A$ is a C$^*$-algebra with the property that every weak-2-local derivation on $A$ is a {\rm(}linear{\rm)} derivation. We prove that every weak-2-local derivation on $C_0(L,A)$ is a {\rm(}linear{\rm)} derivation. Among the consequences we establish that if $B$ is an atomic von Neumann algebra or on a compact 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.