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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: fernando.albiac@unavarra.es)
E. BRIEM
Affiliation:
Science Institute, University of Iceland, 107 Reykjavik, Iceland (email: briem@hi.is)
*
✉For correspondence; e-mail: fernando.albiac@unavarra.es
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Abstract

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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.

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