$\theta$-derivations on convolution algebras

In this paper, we investigate $\theta$-derivations on Banach algebra $L_0^{\mathrm{\infty }}(w{)}^{\ast }$. First, we study the range of them and prove the Singer-Wermer conjucture. We also give a characterization of the space of all $\theta$-derivations on $L_0^{\mathrm{\infty }}(w{)}^{\ast }$. Then, we prove automatic continuity and Posner’s theorems for $\theta$-derivations.