Derivations on Convolution Algebras

Abstract

Abstract. In this paper, we investigate derivations on the noncommutative Banach algebra L ∞ 0 (ω) * equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra L ∞ 0 (ω) * . We then show that a derivation on L ∞ 0 (ω) * is continuous if and only if its restriction to rad(L ∞ 0 (ω) * ) is continuous. We also prove that there is no nonzero centralizing derivation on L ∞ 0 (ω) * . Finally, we prove that the space of all inner derivations of L ∞ 0 (ω) * is continuously homomorphic to the space L ∞ 0 (ω) * /L 1 (ω).

Jordan derivations on the $θ-$Lau products of Banach algebras

“…From this and ( 14) we infer that (θ − φ)(y)d A (a) = 0 (16) for all a ∈ A and y ∈ B. This together with (13) and (15) shows that 16) and ( 17), we get…”

“…Here a question arises: when dose the converse hold? In 1957, Herstein [10] proved that every Jordan derivation on a 2-torsion free prime ring is a derivation; see also [1,5,14,15]. Bresar [4] gave a generalization of Herstein's result for semiprime rings.…”