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.

When you are only interested in

Thus, the category of locally compact totally disconnected Hausdorff spaces , you have is equivalent to assume that the category of commutative AF-algebras; the inverse functors are $A$ is 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(X)$C_0(X,\mathbb{C})$ 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 ...

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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.

When you are only interested in compact spaces, you have to assume that $A$ is unital.

For example, if $X$ is the Cantor set, then it is easy to see that $C(X)$ 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).