The "right" $C^*$ algebraic proof of Bott Periodicity

Question

In learning about the K-theory of $C^*$-algebras, I have encountered the following 3 proofs of Bott periodicity:

$\bullet$ An argument based on Moyal quantization found in "Elements of Noncommutative Geometry."

$\bullet$ An argument based on Toeplitz algebras in Murphy's book.

$\bullet$ A seemingly brute force (but elementary) argument in Rordam, Larsen and Laustsen's book where they prove a variety of density results about projections in matrix algebras.

Which of these proofs is the most "geometrical" in the sense that it has a nice geometric interpretation when we restrict our attention to commutative $C^*$-algebras? As a student interested in noncommutative topology and geometry, if I wanted to study one of these proofs in gory detail, which should it be?

If there are other proofs that are more geometrical or more essential for understanding the phenomenon of Bott periodicity, I would be happy to hear about those too.

Answer 1

I opine that everyones favorite should be the proof using Toeplitz operators (as described in Higson-Roe); essentially due to Atiyah (in ''Bott periodicity and the index of elliptic operators''). In the commutative case, it is the cleanest and most memorable proof. It gives an explicit homotopy inverse to the Bott map as a map $\Omega U \to Z \times BU$ spaces). It is fairly elementary. It is directly linked to the Toeplitz index theorem (that relates the most elementary topological invariant, the winding number, to an index). The Toeplitz index theorem is, moreover, one of the building blocks of the Atiyah-Singer index theorem. The proof can be generalized to the Real case. It sets the stage for Cuntz' proof of periodicity in $KK$-theory. There is one drawback that I am willing to take serious: there is, as far as I can see, no straightforward generalization to the Clifford-linear case.

Answer 2

Probably the argument that you're looking for is based on the "Dirac / Dual Dirac" method. The idea is to exploit the product structure in KK-theory as much as possible - this makes the proof functorial in practically every possible way, and it suggests lots of generalizations (for instance, a variation on the argument can prove the Baum-Connes and analytic Novikov conjectures in a lot of cases). Here's the structure of the argument.

First, construct a "Dirac class" $\delta$ in the K-homology group $K^1(S\mathbb{C})$ and a "dual Dirac class" $\beta$ in the K-theory group $K_1(S\mathbb{C})$ ($S$ means suspension). The Dirac class is the K-homology class of the Dirac operator on $S^1$, or alternatively the K-homology class of the Toeplitz extension. The dual Dirac class is also known as the Bott class (because of it's role in this proof); it can be viewed as the K-theory class of the Clifford multiplication operator.

Second, prove that the map $K_0(\mathbb{C}) \to K_1(S\mathbb{C})$ given by multiplication by $\beta$ and the map $K_1(S\mathbb{C}) \to K_0(\mathbb{C})$ given by multiplication by $\delta$ are inverses and hence both are isomorphisms. This is a direct calculation, and it's really the meat of the proof. It can be reduced to the Toeplitz index theorem, for instance.

Third, simply observe that the map $K_0(A) \to K_1(SA)$ given by multiplication by $\beta$ is an isomorphism by naturality properties of K-theory products.

This is all worked out in a number of places; the expository paper "Group C*-algebras and K-theory" by Guentner and Higson or the textbook "Analytic K-homology" by Higson and Roe both provide this argument in some form, though both references demand some effort to get to the finish line. The first reference might be my favorite; as an added advantage, you'll learn the basics of E-theory along the way.

Answer 3

There's a proof using the Heisenberg group due to Elliott, Natsume, and Nest (The Heisenberg group and K-theory, K-theory 7 (1993), 409-428). But I like the proof in Wegge-Olsen's book, which is based on the original commutative proof by Atiyah.