A Noncommutative Proof of Bott Periodicity

Bott periodicity in K-theory is a rather mysterious object. The classical proofs typically consist of showing that the unitary groups form an Ω-spectrum from which to get a cohomology theory; then showing that that theory is K-theory; and most formidably showing that U(n) is homotopic to U(n+ 2) for all n. However, Cuntz showed that Bott periodicity can be derived in a much simpler way if one resorts to using not only traditional topological spaces, but also ”non-commutative topological spaces,” i.e. C∗-algebras. That in fact, Bott periodicity is not only reflected in the collective topology of the unitary groups, but also encoded in the structure of the Toplitz C∗-algebra. In next two sections we will overview the essential ingredients of non-communitive topology that are needed to follow Cuntz’s argument. Namely the analogues of topological spaces, continuous functions, suspensions, and vector bundles. These two sections are rather sketchy, without any actual proofs given. In the third section we will use the functorial properties of K0 to compute the K-theory of the Toplitz C∗-algebra, and with this we will be able to give Cuntz’s quick proof of Bott periodicity. It should be noted that Cuntz’s proof does not actually even use the definition of K0, just a couple of functorial properties. So the curious reader can skip to the proof of Bott periodicity without any extra difficulties. Finally, it should also be noted that tensor products are rather ubiquetous in the following. It’s a rather unfortunate fact that in general the tensor product of two C∗-algebras is not uniquely defined. One would like to simply that the algebraic tensor product and complete with respect to a C∗-norm, but in general there is more than one C∗-norm on the algebraic tensor product. However, the tensor product is uniquely defined if at least one of the two factors happens to be nuclear. Examples of nuclear C∗-algebras include the commutative ones; the algebra K of compact operators on an infinite dimensional, seperable Hilbert space; and the Toplitz operators. Luckily, in all of the tensor products below at least one of the factors is nuclear. Also, there is good side to C∗-algebra tensor products. As long as at least one of the factors in each tensor product is nuclear, it’s a fact that taking the tensor product of a short exact sequence with a fixed C∗-algebra yields another short exact sequence (i.e., there are no Tor terms). This will come in handy at one point below. The interested reader will find a very nice appendix on tensor products in Wegge-Olsen’s K − Theory and C∗ − Algebras. All material for this paper can be found in Arveson’s A Short Course in Spectral Theory or Wegge-Olsen’s K − Theory and C∗ − Algebras.