The K-inductive structure of the noncommutative Fourier transform

Abstract

The noncommutative Fourier transform $\sigma (U)=V^{-1}$, $\sigma (V)=U$ of the irrational rotation C*-algebra $A_\theta $ (generated by canonical unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta } UV$) is shown to have the following K-inductive structure (for a concrete class of irrational parameters, containing dense $G_\delta $'s). There are approximately central matrix projections $e_1$, $e_2$, $f$ that are σ-invariant and which form a partition of unity in $K_0$ of the fixed-point orbifold $A_\theta ^\sigma $, where $f$ has the form $f = g+\sigma (g) +\sigma ^2(g)+\sigma ^3(g)$, and where $g$ is an approximately central matrix projection as well.