Mohammad Javad Mehdipour and Zahra Saeedi Shiraz University of Technology, Shiraz University of Technology

Abstract : In this paper, we investigate derivations on the noncommutative Banach algebra $L_0^\infty (\omega )^\ast$ equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra $L_0^\infty (\omega )^\ast$. We then show that a derivation on $L_0^\infty (\omega )^\ast$ is continuous if and only if its restriction to $\hbox{rad}(L_0^\infty(\omega )^\ast)$ is continuous. We also prove that there is no nonzero centralizing derivation on $L_0^\infty (\omega )^\ast$. Finally, we prove that the space of all inner derivations of $L_0^\infty (\omega )^\ast$ is continuously homomorphic to the space $L_0^\infty (\omega )^\ast/L^1(\omega)$.