Left $θ$-derivations on weighted convolution algebras

Abstract

Let $θ$ be a homomorphism on $L_0^\infty({\Bbb R}^+, ω)^*$. In this paper, we study left $θ$-derivations on $L_0^\infty({\Bbb R}^+, ω)^*$. We show that every left $θ$-derivation on $L_0^\infty({\Bbb R}^+, ω)^*$ is always a $θ$-derivation, and if $θ$ is isomorphism, then $L_0^\infty({\Bbb R}^+, ω)^*$ has no non-zero left $θ$-derivation. We also investigate automatic continuity, Singer-Wermer's conjecture and Posner's first theorem for left $θ-$derivations on $L_0^\infty({\Bbb R}^+, ω)^*$.