There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
Let $X$ be a topological space. I can think of at least four rings of continuous functions to attach to $X$:
- $C(X)$ is the ring of continuous functions to $\mathbb R$.
- $C_b(X)$ is the ring of bounded functions to $\mathbb R$.
- $C_0(X)$ is the ring of continuous functions that "vanish at infinity" in the following sense: $f\in C_0(X)$ iff for every $\epsilon>0$, there is a compact subset $K \subseteq X$ such that $\left|f(x)\right| < \epsilon$ when $x \not\in K$.
- The ring $C_c(X)$ of functions with compact support.
Option 2. is not very good. For example, it cannot distinguish between a space and its Stone-Cech compactification. My question is what kinds of spaces are distinguished by the others.
For example, I learned from this question that MaxSpec of $C(X)$ is the Stone-Cech compactification of $X$. But it seems that $C(X)$ knows a bit more, in that $C(X)$ has residue fields that are not $\mathbb R$ if $X$ is not compact?
On the other hand, my memory from my C*-algebras class (which was a few years ago and to which I didn't attend well), is that for I think locally-compact Hausdorff spaces $C_0(X)$ knows precisely the space $X$?
So, MOers, what are the exact statements? And if I believe that the correct notion of "space" is "algebra of observables", are there good arguments for preferring one of these algebras (or one I haven't listed) over the others?