Derivations on group algebras of a locally compact abelian group

Abstract

Let G be a locally compact abelian group. In this paper, we study derivations on the Banach algebra $L_0^\infty (G)^*$. We prove that any derivation on $L_0^\infty (G)^*$ maps it into its radical and a derivation on $L_0^\infty (G)^*$ is continuous if and only if its restriction to the right annihilator of $L_0^\infty (G)^*$ is continuous. We also show that the composition of two derivations on $L_0^\infty (G)^*$ is always a derivation on it and the zero map is the only centralizing derivation on $L_0^\infty (G)^*$. Finally, we characterize the space of inner derivations of $L_0^\infty (G)^*$ and show that G is discrete if and only if there exist $i, j, k\in {\mathbb {N}}$ such that $[d(m), n]_i^j=[m, n]_k$ for all $m, n\in L_0^\infty (G)^*$; or equivalently, any inner derivation on $L_0^\infty (G)^*$ is zero.