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

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.