Ideals in $C_B(X)$ arising from ideals in $X$
Tom 245 / 2019
Let $X$ be a completely regular topological space. We assign to each (set-theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded scalar-valued mappings on $X$ equipped with the supremum norm. We then prove several representation theorems for those ideals. This is done by associating a certain subspace of the Stone–Čech compactification $\beta X$ to each ideal of $X$. This subspace has a simple representation, and when the ideal is closed, coincides with its spectrum as a Banach algebra. This in particular provides information about the spectrum of those closed ideals of $C_B(X)$ which have such representations. This includes non-vanishing closed ideals of $C_B(X)$ whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of $C_B(X)$ by relating it to the topological properties of their spectrums. This is illustrated by various examples. Our approach is rather topological and makes use of the theory of Stone–Čech compactification.