Projective Modules over the Non-Commutative Sphere

• Published in: Journal of the London Mathematical Society 1995-06, Vol.51 (3), p.589-602
• Main Author: Walters, Samuel G.
• Format: Article
• Language: English
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms/51.3.589

The positive cone of the K0-group of the non-commutative sphere Bθ is explicitly determined by means of the four basic unbounded trace functionals discovered by Bratteli, Elliott, Evans and Kishimoto. The C*-algebra Bθ is the crossed product Aθ × Ф Z2 of the irrational rotation algebra Aθ by the flip automorphism Ф defined on the canonical unitary generators U, V by Ф(U) = U*, Ф(V) = V*, where VU = e2πiθ UV and θ is an irrational real number. This result combined with Rieffel's cancellation techniques is used to show that cancellation holds for all finitely generated projective modules over Bθ. Subsequently, these modules are determined up to isomorphism as finite direct sums of basic modules. It also follows that two projections p and q in a matrix algebra over Bθ are unitarily equivalent if, and only if, their vector traces are equal: T→[p] = T→[q]. These results will have the following ramifications. They are used (elsewhere) to show that the flip automorphism on Aθ is an inductive limit automorphism with respect to the basic building block construction of Elliott and Evans for the irrational rotation algebra. This will, in turn, yield a two-tower proof of the fact that Bθ is approximately finite dimensional, first proved by Bratteli and Kishimoto.