Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

Question

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which quality of compactness causes that $C(\Omega)$ is a C*-algebra or not. Thanks in advance.

Answer 1

If you start with any topological space $\Omega $ then $ C_b (\Omega) $, the set of bounded continuous functions on $\Omega $, is a C $^*$-algebra.

But the Gelfand transform allows you to show that there exists a locally compact $\Omega'$ with $ C_b (\Omega)\simeq C_b (\Omega') $. So when talking in abstract, you gain nothing by considering non-locally-compact $\Omega $.

As Harald mentioned, when $\Omega $ is compact, all continuous functions on it are bounded. So in such case we can simply talk about $ C(\Omega) $ instead.