Injective hyperbolicity
Let D be a domain in a complex, locally convex Hausdorff vector space. \textit{K. T. Hahn} [Ann. Pol. Math. 39, 71-81 (1981; Zbl 0476.32031)] has defined a differential pseudometric \(\sigma_ D\) on D, which is contracted by all injective holomorphic maps of D into itself and is therefore invariant under the action of the group Aut D of all holomorphic automorphisms of D. The fact that the definition involves only injective holomorphic maps of the open unit disc \(\Delta\) of \({\mathbb{C}}\) into D implies that \(\sigma_ D\) dominates the Kobayashi differential pseudometric \(\kappa_ D\) of D. Thus Kobayashi's hyperbolicity yields Hahn's hyperbolicity. However, it is shown in [loc. cit] that for \({\mathcal E}={\mathbb{C}}\) and \(D={\mathbb{C}}\setminus \{0\}\), \(\sigma_ D\) is a complete differential metric, while - as is well known - \(\kappa_ D\equiv 0.\) The present paper is devoted to investigating further the behaviour of \(\sigma_ D\). After extending to \(\sigma_ D\) a result, established for \(\kappa_ D\) in [the author, Aspects of math. and its appl. 859-871 (1986; Zbl 0617.46049)], whereby the complex Banach spaces are the only complex locally convex Hausdorff vector spaces containing non-empty hyperbolic domains, it will be shown that the complex field is the only unital complex Banach algebra whose group of invertible elements has a connected component which is Hahn-hyperbolic. This result, coupled with the fact that Hahn's pseudo-metric degenerates at some point of the cartesian product of \({\mathbb{C}}\setminus \{0\}\) and of any domain of complex dimension \(\geq 2\), seems to indicate that Hahn's hyperbolicity disjoint from Kobayashi's hyperbolicity does not appear as often as one would expect.
scientific article; zbMATH DE number 4088293