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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.

flag	asked Aug 29 at 22:02 Celeban 95●3

Jon Bannon · Accepted Answer · 2012-08-29 23:19:53Z UTC

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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.

edited Aug 29 at 23:19

answered Aug 29 at 22:59

Jon Bannon
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It is also a standard exercise in text books. – Bill Johnson Aug 30 at 0:31

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. – Jon Bannon Aug 30 at 0:54

Assumptions on a commutative C*-algebra to get a nice C(X) - space

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