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.
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References
J. B. Conway, A Course in Functional Analysis, Springer, 2000.
L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, 1976.
E. Kaniuth, A Course in Commutative Banach Algebras, Graduate Texts in Mathematics, Springer, 2009.
S. H. Kulkarni and B. V. Limaye, Real Function Algebras, Marcel Dekker, Inc., 1992.
Lambert S., Luttman A.: Generalized strong boundary points and boundaries of families of continuous functions. Mediterr. J. Math. 9, 1–20 (2012)
G. M. Leibowitz, Lectures on Complex Function Algebras, Scott, Foresman and Co., 1970.
J. G. Llanova and J. A. Jaramillo, Homomorphisms between algebras of continuous functions, Canad. J. Math. LXI (1) (1989), 132–162.
C. E. Rickart, General Theory of Banach Algebras, Krieger Pub. Co., 1974.
Stone M.H.: Application of the theory of boolean rings to general topology. Trans. Amer. Math. Soc. 41, 132–162 (1937)
E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Inc., 1971.
S. Willard, General Topology, Dover Publications, Inc., 2004.
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Lambert, S., Lee, K. & Luttman, A. On the Generality of Assuming that a Family of Continuous Functions Separates Points. Mediterr. J. Math. 10, 1917–1935 (2013). https://doi.org/10.1007/s00009-013-0323-8
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DOI: https://doi.org/10.1007/s00009-013-0323-8