The Gleason-Kahane-Zelazko theorem and function algebras

Edoardo Vesentini

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 2, page 87-108
  • ISSN: 1120-6330

Abstract

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A theorem due to A. Gleason, J.-P. Kahane and W. Zelazko characterizes continuous characters within the space of all continuous linear forms of a locally multiplicatively convex, sequentially complete algebra. The present paper applies these results to investigate linear isometries of Banach algebras (with particular attention to normal uniform algebras) and of some locally multiplicatively convex algebras. The locally multiplicatively convex algebra of all holomorphic functions on a domain, will be investigated at the end of the paper.

How to cite

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Vesentini, Edoardo. "The Gleason-Kahane-Zelazko theorem and function algebras." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.2 (2005): 87-108. <http://eudml.org/doc/252279>.

@article{Vesentini2005,
abstract = {A theorem due to A. Gleason, J.-P. Kahane and W. Zelazko characterizes continuous characters within the space of all continuous linear forms of a locally multiplicatively convex, sequentially complete algebra. The present paper applies these results to investigate linear isometries of Banach algebras (with particular attention to normal uniform algebras) and of some locally multiplicatively convex algebras. The locally multiplicatively convex algebra of all holomorphic functions on a domain, will be investigated at the end of the paper.},
author = {Vesentini, Edoardo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Linear isometry; Character; Banach algebra; Locally multiplicatively convex algebra; linear isometry; character; locally multiplicatively convex algebra},
language = {eng},
month = {6},
number = {2},
pages = {87-108},
publisher = {Accademia Nazionale dei Lincei},
title = {The Gleason-Kahane-Zelazko theorem and function algebras},
url = {http://eudml.org/doc/252279},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Vesentini, Edoardo
TI - The Gleason-Kahane-Zelazko theorem and function algebras
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/6//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 2
SP - 87
EP - 108
AB - A theorem due to A. Gleason, J.-P. Kahane and W. Zelazko characterizes continuous characters within the space of all continuous linear forms of a locally multiplicatively convex, sequentially complete algebra. The present paper applies these results to investigate linear isometries of Banach algebras (with particular attention to normal uniform algebras) and of some locally multiplicatively convex algebras. The locally multiplicatively convex algebra of all holomorphic functions on a domain, will be investigated at the end of the paper.
LA - eng
KW - Linear isometry; Character; Banach algebra; Locally multiplicatively convex algebra; linear isometry; character; locally multiplicatively convex algebra
UR - http://eudml.org/doc/252279
ER -

References

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  2. GLEASON, A., A characterization of maximal ideals. J. Analyse Math., 19, 1967, 171-172. Zbl0148.37502MR213878
  3. HOFFMAN, K., Banach spaces of analytic functions. Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962, 217 pp. Zbl0117.34001MR133008
  4. HOLSZTYŃSKI, W., Continuous mappings induced by isometries of spaces of continuous functions. Studia Math., 26, 1966, 133-136. Zbl0156.36903MR193491
  5. HÖRMANDER, L., An introduction to complex analysis in several variables. Van Nostrand, Princeton, N.J., 1966, 208 pp. Zbl0138.06203MR203075
  6. KAHANE, J.-P. - ZELAZKO, W., A characterization of maximal ideals in commutative Banach algebras. Studia Math., 29, 1968, 339-343. Zbl0155.45803MR226408
  7. KAMOWITZ, H. - SCHEINBERG, S., The spectrum of automorphisms of Banach algebras. J. Functional Analysis, 4, 1949, 268-276. Zbl0182.17703MR250075
  8. MARTINEAU, A., Sur les fonctionnelles analytiques et la transformation de Fourier-Borel. J. Analyse Math., 9, 1963, 1-164. Zbl0124.31804MR159220
  9. MICHAEL, E., Locally multiplicatively-convex topological algebras. Mem. Amer. Math. Soc., 11, 1952, 82 pp. Zbl0047.35502MR51444
  10. NAGASAWA, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions. Kodai Math. Sem. Rep., 11, 1959, 182-188. Zbl0166.40002MR121645
  11. RICKART, C.E., General theory of Banach algebras. Van Nostrand, Princeton, N.J., 1960. Zbl0095.09702MR115101
  12. RUDIN, W., Functional analysis. McGraw Hill, New York1973. Zbl0867.46001MR365062
  13. SCHEINBERG, S., Automorphisms of commutative Banach algebras. In: R.C. GUNNING (ed.), Problems in analysis. A symposium in honor of Salomon Bochner. Princeton University Press, Princeton, N.J., 1970, 319-323. Zbl0214.13802MR352989
  14. VESENTINI, E., On the Banach-Stone Theorem. Advances in Math., 112, 1995, 135-146. Zbl0854.46025MR1321671DOI10.1006/aima.1995.1030
  15. VESENTINI, E., Weighted composition operators and the Gleason-Kahane-Zelazko theorem. Advances in Math., 191, 2005, 423-445. Zbl1068.46029MR2103220DOI10.1016/j.aim.2004.03.014
  16. VESENTINI, E., Introduction to continuous semigroups. Scuola Normale Superiore, Pisa2002. Zbl0949.47032MR1736550
  17. ZELAZKO, W., A characterization of multiplicative linear functionals in complex Banach algebras. Studia Math., 29, 1968, 339-343. Zbl0155.45803MR229042

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