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?
Best Answer
I work with infinite dimensional manifolds so am extremely distrustful of anything that requires some sort of compactness condition. Most of the time, it's just too restrictive.
Consider a really nice simple space: the coproduct of a countably infinite number of lines, $\sum_{\mathbb{N}} \mathbb{R}$ (coproduct taken in the category of locally convex topological vector spaces). This has the property that any compact subset is contained in a finite subspace. However, any neighbourhood of the origin has to be absorbing (the union of the scalar multiples of it is the whole space) so there aren't any non-zero continuous functions with compact support. That defenestrates option 4.
Particularly simple functions on an infinite dimensional vector space are the cylinder functions. These are important in measure theory on such spaces. A cylinder function has the property that it factors through a projection to a finite dimensional vector space. Such functions can be continuous and can be bounded, but (apart from the zero function) never vanish at infinity and never have compact support. Thus option 3 joins option 4 in the flowerbed.
As for option 2, I have no particular qualms about it except that it's not stable under partitions-of-unity. Assuming that I have such, then any continuous function can be written as a sum of bounded functions so when doing standard p-of-1 constructions I have to assume that my starting family is uniformly bounded (if that's the right term).
Well, I just remembered one qualm about option 2: if I go up the scale of differentiability then it gets increasingly hard to justify global bounds on the derivatives. I have a memory of John Roe telling me of some result that he'd proved which was to do with bounding all derivatives of a smooth function in some fashion. I don't recall the exact conditions, but the conclusion was that the only functions that satisfied them were trigonometric.
As others have said, if you are really only interested in (locally) compact spaces then the other options have meaning (functionally Hausdorff - points separated by functions - is assumed). But then the title of your question should have been: "Which is the correct ring of functions for a Locally Compact Hausdorff Space?".