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Paper IPM / M / 17796

School of Mathematics
Title:	One-point extensions of a Tychonoff space $X$ via closed ideals of $C_{B}(X)$
Author(s):	Alireza Olfati
Status:	Published
Journal:	Bull. Malaysian Math. Soc.
Vol.:	47
Year:	2024
Pages:	1-27
Supported by:	IPM

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 \v{C}ech-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.

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“School of Mathematics”

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