TITLE: Which is the correct ring of functions for a topological space?
QUESTION [22 upvotes]: 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?

REPLY [5 votes]: If one can distill one central fact from the answers above and elsewhere, then it is that if one wants a good duality theory for algebras of continous functions, one has to move on from Banach spaces or algebras.  This fact was recognised at least 50 years ago, and a suitable generalisation was discovered---the so-called strict topology on the space of bounded, continuous functions on a, say, completely regular space.  Despite the eminence of its discoverers and advocates (Beurling, Herz, Buck), it seems to have become forgotten lore.  A lucid argument on its behalf can be found in the 1960 Transactions paper by Herz, "The spectral theory of bounded functions" (as the title suggests, the motivation for its introduction comes from Harmonic Analysis).  The original version used weighted seminorms and was valid for locally compact spaces.  It was subsequently extended to completely regular spaces by several authors.  The topology can be defined most succinctly as the finest locally convex one which agrees with compact convergence on the unit ball of $C^b(X)$.  It is a complete locally convex spaces, its compact sets and  convergent sequences can be simply characterised (the latter are the uniformly bounded sequences which converge uniformly on compacta). Its dual is the space of bounded Radon measures on $X$, the natural version of the Stone-Weierstra\ss theorem holds for it,  its spectrum is naturally identifiable with $X$ and so one has a version of Gelfand-Naimark theory.  One can also characterise the topological properties of $X$  in terms of those of $C^b(X)$, all in principle, many in practice.  Importantly, one can characterise local compactness of $X$ in terms of the property of the algebra $C^b(X)$.  A full account can be found in the book "Saks Spaces and Applications to Functional Analysis".