Left $\theta$-derivations on weighted convolution algebras

Abstract

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