How I Came to Terms with a Proof in Davidson’s Book

The book C*-Algebras by Example by Ken Davidson is a standard text for C*-algebraists and well-praised. But a proof of one of the theorems bothered me a little. The theorem states that the C*-algebra of continuous functions on the Cantor set is AF (approximately finite). It’s a pretty easy result. The general statement that the C*-algebra of continuous functions on a compact totally disconnected space is AF is not difficult. By considering the nerves of open covers, one can see that a compact Hausdorff space is totally disconnected (which is equivalent to being zero-dimensional in this case) if and only if the space is the inverse limit of spaces with finitely many points. So by taking the contrapositive functor of taking continuous functions, we see that the C*-algebra of continuous functions over our space is the inductive limit of finite-dimensional algebras, and thus AF. A nice clean proof.

But this isn’t the proof that Davidson uses! Instead, he considers the standard “middle-third” picture of the Cantor set and then considers locally constant functions (i.e. constant on each connected component) on each intermediate set. (If my memory serves, this book and most of my books are currently in storage.) The algebras of these locally constant functions are finite-dimensional and the inductive limit of these algebras is the algebra of continuous functions over the Cantor set, and the result is proved. There is nothing technically wrong with the proof, but it didn’t seem to fit into the larger picture. This algebra being AF is exclusively a property of the underlying space being totally disconnected and not of the standard picture. So it was slightly annoying.

But this isn’t as problematic as I first thought. For starters, the middle-thirds picture reflects (refinements of) clopen covers and so this picture does reflect the total disconnectedness of the Cantor set. Also given a finite clopen cover of a compact space, we can identify the (geometric realization of the) nerve of the covering with the quotient space by identifying points within a connected component. As I mentioned yesterday, the quotient corresponds to a subalgebra of the larger algebra, which is the locally constant functions! So in fact, this proof that seemed off was a rephrasing of the proof that I had in mind from the start!


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