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| Paper IPM / M / 17796 |
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| Abstract: | |
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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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