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Analytic structure in fibers

Volume 240 / 2018

Javier Falcó, Domingo García, Manuel Maestre, Richard M. Aron Studia Mathematica 240 (2018), 101-121 MSC: Primary 46B20; Secondary 46B22, 46B25. DOI: 10.4064/sm8601-4-2017 Published online: 22 September 2017

Abstract

Let $B_X$ be the open unit ball of a complex Banach space $X,$ and let $\mathcal H^\infty (B_{X})$ and $\mathcal A_u(B_X)$ be, respectively, the algebra of bounded holomorphic functions on $B_X$ and the subalgebra of uniformly continuous holomorphic functions on $B_X.$ In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of $\mathcal H^\infty (B_X),$ we prove that the fiber in $\mathcal M(\mathcal H^\infty (B_{c_0}))$ over any point of the distinguished boundary of the closed unit ball $\overline {B}_{\ell _{\infty }}$ of $\ell _\infty $ contains an analytic copy of $B_{\ell _{\infty }}$. In the case of $\mathcal A_u(B_X)$ we prove that if there exists a polynomial whose restriction to $B_{X}$ is not weakly continuous at some point, then the fiber over every point of the open unit ball of the bidual contains an analytic copy of $\mathbb {D}.$

Authors

  • Javier FalcóDépartement de Mathématique
    Université de Mons
    20 Place du Parc
    7000 Mons, Belgium
    e-mail
  • Domingo GarcíaDepartamento de Análisis Matemático
    Universidad de Valencia
    Doctor Moliner 50
    46100 Burjasot (Valencia), Spain
    e-mail
  • Manuel MaestreDepartamento de Análisis Matemático
    Universidad de Valencia
    Doctor Moliner 50
    46100 Burjasot (Valencia), Spain
    e-mail
  • Richard M. AronDepartment of Mathematical Sciences
    Kent State University
    Kent, OH 44242, U.S.A.
    e-mail

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