Injective hyperbolicity
type: Item
instance of: scholarly article
title: Injective hyperbolicity
zbMATH Open document ID: 0665.46040
author: Edoardo Vesentini
published in: Ricerche di Matematica
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 32Q45
zbMATH DE Number: 4088293
zbMATH Keywords: locally convex Hausdorff vector space
zbMATH Keywords: differential pseudometric
zbMATH Keywords: Kobayashi differential pseudometric
zbMATH Keywords: non-empty hyperbolic domains
zbMATH Keywords: Hahn's pseudo-metric
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4088293