Journal Article

REAL BANACH ALGEBRAS AS π’ž (𝒦) ALGEBRAS

F. Albiac and E. Briem

inΒ The Quarterly Journal of Mathematics

Volume 63, issue 3, pages 513-524
Published in print September 2012 | ISSN: 0033-5606
Published online April 2011 | e-ISSN: 1464-3847 | DOI:Β https://dx.doi.org/10.1093/qmath/har005
REAL BANACH ALGEBRAS AS π’ž (𝒦) ALGEBRAS

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The classical theorem of Gelfand provides a representation of a commutative complex unital Banach algebra as a subalgebra of π’žβ„‚(𝒦) of continuous complex-valued functions defined on a compact Hausdorff space π’ž. Since the complex algebras can be regarded as a subclass of the real algebras, it is natural to ask what can be said about this larger class. As it happens, a real commutative Banach algebra π’œ does admit a Gelfand representation a β†’ Γ’ as in the complex case, where each Γ’: 𝒦 β†’ β„‚ is a continuous function. However, if we attempt to represent a commutative real Banach algebra as a subalgebra of π’ž(𝒦) of continuous real-valued functions in the same fashion, complications arise and in the general case it need not even be true. In this article, we will look at two conditions on π’œ that will imply that the representation of π’œ as a space of continuous functions consists only of real-valued functions. The methods we use are intrinsic, that is to say, they do not rely on the complexification of the algebra.

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Subjects: Pure Mathematics

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