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

For a Tychonoff space X, let \(C_{B}(X)\) $C_B(X)$ be the \(C^{*}\) $C^{\ast }$ -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 \(C_{B}(X)\) $C_B(X)$ . We show that every closed ideal H of \(C_{B}(X)\) $C_B(X)$ produces a Tychonoff one-point extension \(X(\infty _{H})\) $X({\infty }_H)$ 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 \(C_{B}(X)\) $C_B(X)$ is isometrically \(*\) $\ast$ -isomorphic with the \(C^{*}\) $C^{\ast }$ -algebra \(C_{B}\left( X(\infty _{H})\right) \) $C_B\left(X,(,{\infty }_H,)\right)$ . We provide a description for the Čech–Stone compactification of an arbitrary Tychonoff one-point extension of X as a quotient space of \(\beta X\) $\beta X$ via a closed ideal of \(C_{B}(X)\) $C_B(X)$ . Then, we establish a characterization of closed ideals of \(C_{B}(X)\) $C_B(X)$ that have countable topological generators. Finally, an intrinsic characterization of the multiplier algebra of an arbitrary closed ideal of \(C_{B}(X)\) $C_B(X)$ is given.