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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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  • 1. Barton, T.J., Friedman, Y.: Bounded derivations of $\text{JB}^{\ast }$-triples. Quart. J. Math. Oxford 41, 255-268 (1990). MR 91j:46086
  • 2. Barton, T.J., Timoney, R. M.: Weak$^{*}$-continuity of Jordan triple products and applications, Math. Scand. 59, 177-191 (1986). MR 88d:46129
  • 3. Behrends, E.: M-Structure and the Banach-Stone theorem, Lecture Notes in Mathematics Vol. 736. Berlin-Heidelberg-New York: Springer Berlin-Heidelberg-New York: Springer 1979. MR 81b:46002
  • 4. Braun, R. , Kaup, W. , Upmeier, H.: On the automorphisms of circular and Reinhardt domains in complex Banach spaces, Manuscripta Math. 25, 97-133 (1978). MR 80g:32003
  • 5. Dineen, S., Timoney, R.M.: The centroid of a $\text{JB}^{\ast }$-triple system. Math. Scand. 62, 327-342 (1988). MR 89j:46067
  • 6. Harris, L.A.: A generalization of C$^{*}$-algebras. Proc. London Math. Soc. 42, 331-361 (1981). MR 82e:46089
  • 7. Harris, L.A., Kaup, W.: Linear algebraic groups in infinite dimensions. Illinois J. Math. 21, 666-674 (1977). MR 57:544
  • 8. Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503-529 (1983). MR 85c:46040
  • 9. Mellon, P.: Dual manifolds of $\text{JB}^{\ast }$-triples of the form ${\mathcal{C}}(X, \, U)$. Proc. R. Ir. Acad. 93 A, 27-42 (1993). MR 94k:58011
  • 10. Mellon, P.: Symmetric manifolds of compact type associated to the $\text{JB}^{\ast }$-triples ${\mathcal{C}}_{0}(X,\,Z)$, Math. Scand. 78, 19-36 (1996). MR 97h:46112
  • 11. Upmeier, H.: Symmetric Banach manifolds and Jordan C$^{*}$-algebras, North Holland Math. Studies 104 North-Holland, Amsterdam 1985. MR 87a:58022

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Additional Information

J. M. Isidro
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago, Santiago de Compostela, Spain

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

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