Holomorphic injective extensions of functions in P(K) and algebra generators |
| |
Authors: | Raymond Mortini |
| |
Affiliation: | 1.Université de Lorraine,Département de Mathématiques et Institut élie Cartan de Lorraine, UMR 7502 Ile du Saulcy,Metz,France |
| |
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 K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ?1 ? P(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 f ∈ P(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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|