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I have the following question,

Is it possible to get somehow a compact Hausdorff space $X$ which is second-countable from a unital commutative C*-algebra. If it is possible, what should we assume for our C*-algebra. Gelfand-Naimark theorem gives us $C(X)$, where $X$ is a compact Hausdorff space, but I'm asking how to get it with the second axiom of countability. Thank you in advance for any answers or hints.

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The space $X$ is second countable if and only if $C(X)$ is separable for the norm. This is proved, for example, as Theorem 2.4 of the little article "Notes on the Separability of C* algebras" by Chun-Yen Chou. Actually, the short proof given there works also for locally compact Hausdorff spaces and therefore non-unital $C^{*}$-algebras.

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    $\begingroup$ It is also a standard exercise in text books. $\endgroup$ – Bill Johnson Aug 30 '12 at 0:31
  • $\begingroup$ You're right Bill. This was probably best left as a long comment, should the OP have liked to move the question to math.stackexchange. $\endgroup$ – Jon Bannon Aug 30 '12 at 0:54

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