Holomorphic automorphisms of continuous products of balls
Let \({\mathcal C}(\omega,E)\) denote the space of all continuous \(E\)-valued functions on \(\omega\), where \(\omega\) is a compact Hausdorff topological space and \(E\) a complex Banach space. The authors study the group of holomorphic automorphisms of continuous product domains \(D\) in \({\mathcal C} (\omega,E)\). These product domains have natural fibrations and it is of interest to understand the fiber preserving automorphisms. Denote by \(G(D)\) the identity component in the group of all holomorphic automorphisms of \(D\). If \((D_w)_{w \in\omega}\) is a family of domains in \(E\) with a mild restriction on smoothness of the boundary and some other mild condition, then for \(D=\{f\in {\mathcal C} (\omega, E):f(w)\in D_w, w\in\omega\}\) the authors show that all transformations in \(G(D)\) are fiber preserving. In the case of continuous families of balls in \(JB^*\)-triples the authors characterize the complete vector fields in the product domain reobtaining old results and also showing that others are not longer valid in the new context by providing counterexamples.
scientific article; zbMATH DE number 1515374