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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Behaviour of holomorphic automorphisms
on equicontinuous subsets of the space ${\mathcal{C}} (\protect\Omega ,E)$


Author: J. M. Isidro
Journal: Proc. Amer. Math. Soc. 127 (1999), 437-446
MSC (1991): Primary 46G20, 22E65
MathSciNet review: 1469414
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Abstract: Consider a compact Hausdorff topological space $\Omega $, a $\text{JB}^{\ast }$-triple $E$ and $F: = {\mathcal{C}}(\Omega , \, E)$, the $\text{JB}^{\ast }$-triple of all continuous $E$-valued functions $f\colon \Omega \to E$ with the pointwise operations and the norm of the supremum. Let ${\mathsf{G}}$ be the group of all holomorphic automorphisms of the unit ball $B_{F}$ of $F$ that map every equicontinuous subset lying strictly inside $B_{F}$ into another such a set. The real Banach-Lie group ${\mathsf{G}}$ and its Lie algebra are investigated. The identity connected component of ${\mathsf{G}}$ is identified when $E$ has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case $E={\mathbb{C}}$.


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J. M. Isidro
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago, Santiago de Compostela, Spain
Email: jmisidro@zmat.usc.es

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04585-2
PII: S 0002-9939(99)04585-2
Received by editor(s): November 7, 1996
Received by editor(s) in revised form: May 19, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society