Bott periodicity and almost commuting matrices

We give a proof of the Bott periodicity theorem for topological K K -theory of C ∗ C^* -algebras based on Loring’s treatment of Voiculescu’s almost commuting matrices and Atiyah’s rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu’s localization algebra and an associated explicit formula for the pairing between the first K K -homology and first K K -theory groups of a (separable) C ∗ C^* -algebra.