On dual-valued operators on Banach algebras (Q713061)

Let \(D\) be a bounded derivation from a regular Banach algebra \(A\) to \(A^*\). The paper considers under which condition the transpose \(D^*\) of \(D\) becomes a bounded derivation on \(A^{**}\). The authors restrict their attention to the class of bounded derivations \(D: A \to A^*\) such that \(\langle a, D(a) \rangle= 0\) for all \(a \in A\). An application to Beurling algebras on the additive group of \(\mathbb Z\) is given. Conditions under which the second transpose of an \(A^*\)-valued bounded derivation on \(A\) becomes a bounded derivation on \(A^{**}\) endowed with the first Arens product were investigated by \textit{H. G. Dales}, \textit{A. Rodríguez-Palacios} and \textit{M. V. Velasco} [J. Lond. Math. Soc., II. Ser. 64, No. 3, 707--721 (2001; Zbl 1023.46051)] and \textit{S. Barootkoob} and \textit{H. R. E. Vishki} [Bull. Aust. Math. Soc. 83, No. 1, 122--129 (2011; Zbl 1270.46042)].