Given a locally compact Hausdorff space, does $C_0(X)$, the continuous functions vanishing at infinity, determine the topology of $X$?
For example, for a net $\{x_{\alpha}\}\subset X$ if I have $f(x_\alpha)\to f(x)$ for all $f\in C_0(X)$, does it follow that $x_{\alpha} \to x$ ?
I cannot find any reference to the Banach-Stone theorem that proves this. I would really appreciate some feedback.
Thanks!
Best Answer
Yes. If $X$ is a locally compact Hausdorff space, then $C_0(X)$ is an Abelian C*-algebra. Moreover, the Gelfand spectrum of $C_0(X)$ is homeomorphic to $X$.
In another word, if $C_0(X)$ and $C_0(Y)$ are C*-isomorphic, then $X$ and $Y$ are homeomorphic.
(In fact, every Abelian C*-algebra is of the form $C_0(X)$. This is known as the Gelfand Theorem.)