TITLE: The ring of continuous real-valued functions on a Stone space
QUESTION [7 upvotes]: Let $X$ be a topological space and consider the ring $C(X,\mathbb{R})$ of continuous real-valued functions on $X$ (where ring-addition and multiplication are defined in the obvious point-wise way).
It is well-known that this ring uniquely determines the topological space, if the latter is compact and Hausdorff (Gelfand-Duality).
My question is the following: What properties does the ring $C(X,\mathbb{R})$ have, if $X$ is a Stone space (that is, if $X \in Comp_2$ is also totally disconnected)? Ideally, I would like to have a statement along the following line:
$X$ is a Stone space (if and) only if the ring $C(X,\mathbb{R})$ has the property [nice property given to me by the lovely folks from MO].

REPLY [3 votes]: The answer is given by the following theorem (Davidson's $C^\*$-Algebras by Example, III, 2.5).

Let $A$ be a commutative C*-Algebra. Then the following statements are equivalent:

$A$ is an AF-algebra, i.a. a colimit of a sequence of finite-dimensional C*-algebras
$A$ is separable and the projections in $A$ generate a dense subspace.
The spectrum of $A$ is totally disconnected.


Thus, the category of locally compact totally disconnected Hausdorff spaces is equivalent to the category of commutative AF-algebras; the inverse functors are $C_0(-,\mathbb{C})$ and $\mathrm{Spm}$. Under this equivalence, compact corresponds to unital. The finite-dimensional commutative algebras are just powers of $\mathbb{C}$, corresponding to finite sets, but colimits produce interesting examples:
For example, if $X$ is the Cantor set, then it is easy to see that $C_0(X,\mathbb{C})$ is the colimit of the sequence $\mathbb{C} \to \mathbb{C}^2 \to \mathbb{C}^{2^2} \to \dotsc$, where $\mathbb{C}^{2^n} \to \mathbb{C}^{2^{n+1}}, a \mapsto (a,a)$. In general, AF-algebras are classified via so-called Bratteli-diagrams (see loc. cit).
By the way, there is a nice connection to Stone duality, which says that $C_0(-,\mathbb{F}_2)$ exhibits an equivalence of categories between the category of locally compact totally disconnected Hausdorff spaces and the category of boolean rings. In the diagram you just exchange $\mathbb{C}$ with $\mathbb{F}_2$. I would love to see a purely algebraic functor $\mathbb{F}_2 \otimes_{\mathbb{C}} (-)$, which doesn't use the spectrum as an intermediate step ...