A generalization of the Arens extension for Banach algebras
Let \(A\) be a complex Banach algebra and \(H_b(A)\) the Fréchet algebra of all holomorphic functions on \(A\) bounded on bounded sets. The authors introduce a multiplicative convolution \(\star\) in the strong dual space \(H_b(A)^*\) in the following way: Every \(x\in A\) determines a linear operator on \(A\) according to \(m_x(y)=xy\). Then the composition operator \(Q_x\) on \(H_b(A)\) with symbol \(m_x\) leads to its dual operator \(Q_x^*\) on \(H_b(A)^*\). With its help, the multiplicative convolution of \(f\in H_b(A)\) and \(\psi\in H_b(A)^*\) is defined by \((\psi\star f)(x)=Q_x^*(\psi)(f)\). And finally, the multiplicative convolution of \(\varphi,\psi\in H_b(A)^*\) is defined by \((\varphi\star \psi)(f)=\varphi(\psi\star f)\). It is proved that \((\varphi\star \psi)\in H_b(A)^*\) and that \(\star\) is an extension of the Arens product in \(A^{**}\) if one considers \(A^{**}\) as a subset of \(H_b(A)^*\) by identifying every \(x\in A^{**}\) with the evaluation functional \(\tilde{\delta}_x(f)= \tilde{f}(x)\), where \( \tilde{f}\) is the Aron-Berner extension of \(f\).
scientific article; zbMATH DE number 6418454