Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T09:32:36.370Z Has data issue: false hasContentIssue false
  • English
  • Français

REPRESENTATIONS OF REAL BANACH ALGEBRAS

Part of: Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  12 May 2010

F. ALBIAC  and
E. BRIEM
F. ALBIAC*
Affiliation:
Departamento de Matemáticas, Universidad Pública de Navarra, Pamplona 31006, Spain (email: [email protected])
E. BRIEM
Affiliation:
Science Institute, University of Iceland, 107 Reykjavik, Iceland (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Keywords

commutative real Banach algebra𝒞(K)-spacemaximal ideal spacerepresentation of algebras

MSC classification

Secondary: 46J10: Banach algebras of continuous functions, function algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 289 - 300
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first-named author acknowledges support from the Spanish Ministerio de Ciencia e Innovación Research Project Operadores, retículos, y geometría de espacios de Banach, reference number MTM2008-02652/MTM.

References

[1]Albiac, F. and Kalton, N. J., Topics in Banach Space Theory, Graduate Texts in Mathematics, 233 (Springer, New York, 2006).Google Scholar
[2]Albiac, F. and Kalton, N. J., ‘A characterization of real 𝒞(K)-spaces’, Amer. Math. Monthly 114(8) (2007), 737–743.CrossRefGoogle Scholar
[3]Arens, R., ‘Representation of *-algebras’, Duke Math. J. 14 (1947), 269–282.CrossRefGoogle Scholar
[4]Bonsall, F. F. and Duncan, J., Complete Normed Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80 (Springer, New York, 1973).CrossRefGoogle Scholar
[5]Doran, R. S. and Belfi, V. A., Characterizations of C*-Algebras, Monographs and Textbooks in Pure and Applied Mathematics, 101 (Marcel Dekker Inc., New York, 1986).Google Scholar
[6]Gelfand, I. and Neumark, M., ‘On the imbedding of normed rings into the ring of operators in Hilbert space’, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197–213 (English, with Russian summary).Google Scholar
[7]Goodearl, K. R., Notes on Real and Complex C*-Algebras, Shiva Mathematics Series, 5 (Shiva Publishing Ltd., Nantwich, 1982).Google Scholar
[8]Katznelson, Y., ‘Sur les algèbres dont les éléments non négatifs admettent des racines carrées’, Ann. Sci. École Norm. Sup. (3) 77 (1960), 167–174 (in French).CrossRefGoogle Scholar
[9]Kulkarni, S. H. and Limaye, B. V., ‘Gel’fand-Naimark theorems for real Banach *-algebras’, Math. Japon. 25(5) (1980), 545–558.Google Scholar
[10]Kulkarni, S. H. and Limaye, B. V., Real Function Algebras, Monographs and Textbooks in Pure and Applied Mathematics, 168 (Marcel Dekker, New York, 1992).Google Scholar
[11]Palmer, T. W., Banach Algebras and the General Theory of *-Algebras, Vol. I, Encyclopedia of Mathematics and its Applications, 49 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar