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Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 51–69 | Cite as

Holomorphic injective extensions of functions in P(K) and algebra generators

  • Raymond Mortini
Article
  • 45 Downloads

Abstract

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta KC and fP(K) with the property that fP(K) is a homeomorphism of K onto its image, but for which f −1P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|G of the Gelfand-transform f* of an injective function fP(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.

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Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  1. 1.Université de LorraineDépartement de Mathématiques et Institut Élie Cartan de Lorraine, UMR 7502 Ile du SaulcyMetzFrance

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