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

For an algebra A of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that A separates points in the sense that for each distinct pair x, y ∈ X, there exists an f ∈ A such that f(x) 6= f(y). If A does not separate points, it is known that there exists an algebra  on a compact Hausdorff space (X̂, τ̂) that does separate points such that the map A 7→  is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume A separates points. The construction of  and (X̂, τ̂) does not require that A has any algebraic structure nor that (X, τ) 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 A of bounded, complex-valued, continuous functions on any topological space (X, τ). We also demonstrate that further structures may be preserved by the mapping A 7→ Â, 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.