Abstract
Let [Σ, a] be an arbitrary pair and consider a homomorphism φ: a → â(φ) of a onto ℂ. By Proposition 3.1, cp will be continuous iff there exists a compact set K ⊂⊂ Σ that dominates φ; i.e.
In general the dominating compact set K will not be uniquely determined. For example any larger compact set will also serve. Denote by Kφ the collection of all compact subsets of Σ that dominate φ. A set K0 ∈ Kφ is called a support for φ if it is minimal; i.e. no compact set properly contained in K0 dominates φ. It is easy to prove by a routine application of Zorn’s lemma that each element of Kφ contains a support for φ. Note that φ may be supported by many distinct compact sets. For example in [ℂ, ℘] a point evaluation is supported by every circle that contains the point in its interior.
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© 1979 Springer-Verlag New York Inc.
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Rickart, C.E. (1979). The Šilov Boundary and Local Maximum Principle. In: Natural Function Algebras. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8070-2_3
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DOI: https://doi.org/10.1007/978-1-4613-8070-2_3
Publisher Name: Springer, New York, NY
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