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

- 82 Downloads

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J. B. Conway,
*A Course in Functional Analysis*, Springer, 2000.Google Scholar - 2.L. Gillman and M. Jerison,
*Rings of Continuous Functions*, Springer-Verlag, 1976.Google Scholar - 3.E. Kaniuth,
*A Course in Commutative Banach Algebras*, Graduate Texts in Mathematics, Springer, 2009.Google Scholar - 4.S. H. Kulkarni and B. V. Limaye,
*Real Function Algebras*, Marcel Dekker, Inc., 1992.Google Scholar - 5.Lambert S., Luttman A.: Generalized strong boundary points and boundaries of families of continuous functions. Mediterr. J. Math.
**9**, 1–20 (2012)MathSciNetCrossRefGoogle Scholar - 6.G. M. Leibowitz,
*Lectures on Complex Function Algebras*, Scott, Foresman and Co., 1970.Google Scholar - 7.J. G. Llanova and J. A. Jaramillo,
*Homomorphisms between algebras of continuous functions*, Canad. J. Math.**LXI**(1) (1989), 132–162.Google Scholar - 8.C. E. Rickart,
*General Theory of Banach Algebras*, Krieger Pub. Co., 1974.Google Scholar - 9.Stone M.H.: Application of the theory of boolean rings to general topology. Trans. Amer. Math. Soc.
**41**, 132–162 (1937)CrossRefGoogle Scholar - 10.E. L. Stout,
*The Theory of Uniform Algebras*, Bogden and Quigley, Inc., 1971.Google Scholar - 11.S. Willard,
*General Topology*, Dover Publications, Inc., 2004.Google Scholar