The maximal ideal space of a commutative $C^*$-algebra is a compact Hausdorff space and it is one way to construct a compactification of a topological space. I am looking for an introductory book in commutative $C^*$-algebra that has a treatment of the properties of a topological space in the sense that the book gives properties of a topological space that can be derived from the $C^*$-algebra of the continuous bounded complex-valued function on that space and vice versa.

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    $\begingroup$ An older book (1971), my I studied it when I still had access to the university library : Semadeni, Banach spaces of continuous functions (vol 1), not sure if a vol. 2 ever came out. Quite thorough on the basics, in my memory. $\endgroup$ – Henno Brandsma Apr 7 at 21:25

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