# Bott periodicity and almost commuting matrices

## TL;DR

This paper provides a proof of Bott periodicity in topological K-theory of C*-algebras using almost commuting matrices, Atiyah's rotation trick, and connections to the Dirac operator on the circle.

## Contribution

It introduces a novel proof of Bott periodicity based on Loring's approach and relates it to the Dirac operator via localization algebra and explicit pairings.

## Key findings

- Proof of Bott periodicity using almost commuting matrices
- Connection between K-theory and Dirac operator on the circle
- Explicit formula for pairing between K-homology and K-theory groups

## Abstract

We give a proof of the Bott periodicity theorem for topological K-theory of 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-homology and first K-theory groups of a (separable) C*-algebra.