On bounded Dual-valued derivations on certain Banach algebras

A. L.,Barrenechea, C. C.,Peña

We consider the class $\mathfrak{D}(\mathcal{U})$of bounded derivations $\mathcal{U}\overset{d}\to\mathcal{U}^*$defined on a Banach algebra $\mathcal{U}$ with values in its dual space $\mathcal{U}^*$so that $\langle x,d(x)\rangle =0$ for all $x\in \mathcal{U}$.The existence of such derivations is shown, but lacking the simplest structure of an inner one.We characterize the elements of $\mathfrak{D}(\mathcal{U})$if $\operatorname{span}(\mathcal{U}^2)$ is dense in $\mathcal{U}$or if $\mathcal{U}$ is a unitary dual Banach algebra.