Mediterranean Journal of Mathematics

, Volume 10, Issue 4, pp 1917–1935 | Cite as

On the Generality of Assuming that a Family of Continuous Functions Separates Points

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Abstract

For an algebra \({\mathcal{A}}\) of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that \({\mathcal{A}}\) separates points in the sense that for each distinct pair \({x, y \in X}\) , there exists an \({f \in \mathcal{A}}\) such that \({f(x) \neq f(y)}\) . If \({\mathcal{A}}\) does not separate points, it is known that there exists an algebra \({\widehat{\mathcal{A}}}\) on a compact Hausdorff space \({(\widehat{X}, \widehat{\tau})}\) that does separate points such that the map \({\mathcal{A} \mapsto \widehat{\mathcal{A}}}\) is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume \({\mathcal{A}}\) separates points. The construction of \({{\widehat{\mathcal{A}}}}\) and \({(\widehat{X}, \widehat{\tau})}\) does not require that \({\mathcal{A}}\) has any algebraic structure nor that \({(X, \tau)}\) has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family \({\mathcal{A}}\) of bounded, complex-valued, continuous functions on any topological space \({(X, \tau)}\) . We also demonstrate that further structures may be preserved by the mapping \({\mathcal{A} \mapsto \widehat{\mathcal{A}}}\) , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.

Mathematics Subject Classification (2010)

Primary 46E25 46J10 Secondary 46J20 46E15 

Keywords

Separation of points Weak peak points Lipschitz algebra Real function algebra 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsColby CollegeWatervilleUSA
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Defense Experimentation and Stockpile StewardshipNational Security TechnologiesLas VegasUSA

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