Some results on second transpose of a dual valued derivation

Abstract

Let $A$ be a Banach algebra and $X$ be an arbitrary Banach $A$-module. In this paper, we study the second transpose of derivations with value in dual Banach $A$-module $X^{*}.$ Indeed, for a continuous derivation $D:Alongrightarrow X^{*}$ we obtain a necessary and sufficient condition such that the bounded linear map $Lambdacirc D^{primeprime}:A^{**}longrightarrow X^{***}$ to be a derivation, where $Lambda$ is composition of restriction and canonical injection maps. This characterization generalizes some well known results in [2].