One-Point Extensions of a Tychonoff Space X via Closed Ideals of $$C_{B}(X)$$

Abstract

For a Tychonoff space X, let \(C_{B}(X)\) be the \(C^{*}\)-algebra of all bounded complex-valued continuous functions on X. In this paper, we mainly discuss Tychonoff one-point extensions of X arising from closed ideals of \(C_{B}(X)\). We show that every closed ideal H of \(C_{B}(X)\) produces a Tychonoff one-point extension \(X(\infty _{H})\) of X. Moreover, every Tychonoff one-point extension of X can be obtained in this way. As an application, we study the partially ordered set of all Tychonoff one-point extensions of X. It is shown that the minimal unitization of a non-vanishing closed ideal H of \(C_{B}(X)\) is isometrically \(*\)-isomorphic with the \(C^{*}\)-algebra \(C_{B}\left( X(\infty _{H})\right) \). We provide a description for the Čech–Stone compactification of an arbitrary Tychonoff one-point extension of X as a quotient space of \(\beta X\) via a closed ideal of \(C_{B}(X)\). Then, we establish a characterization of closed ideals of \(C_{B}(X)\) that have countable topological generators. Finally, an intrinsic characterization of the multiplier algebra of an arbitrary closed ideal of \(C_{B}(X)\) is given.