ON BOUNDED DUAL-VALUED DERIVATIONS ON CERTAIN BANACH ALGEBRAS

A. L. Barrenechea and C. C. Pena

Abstract: 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.

Keywords: Dual Banach algebras, approximation property, dual Banach pairs, nuclear operators, shrinking basis and associated sequence of coefficient functionals

Classification (MSC2000): 46H35; 47D30

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