Chern Characters of Fourier Modules

Let Aθ denote the rotation algebra—the universal C*-algebra generated by unitaries U, V satisfying VU = e2πiθUV, where θ is a fixed real number. Let σ denote the Fourier automorphism of Aθ defined by U ↦ V, V ↦ U-1, and let denote the associated C*-crossed product. It is shown that there is a canonical inclusion for each θ given by nine canonical modules. The unbounded trace functionals of Bθ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero HC0(Bθ) when θ is irrational. The Chern characters of the nine modules—and more importantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi’s theta functions. Also derived are explicit equations connecting unbounded traces across strongMorita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense Gδ set of values of θ one has and is generated by the nine classes constructed here.