Projective Modules over the Non‐Commutative Sphere

The positive cone of the K 0 ‐group of the non‐commutative sphere B θ is explicitly determined by means of the four basic unbounded trace functionals discovered by Bratteli, Elliott, Evans and Kishimoto. The C∗‐algebra B θ is the crossed product A θ × Ф Z 2 of the irrational rotation algebra A θ by the flip automorphism Ф defined on the canonical unitary generators U, V by Ф( U ) = U ∗, Ф( V ) = V ∗, where VU = e 2πiθ 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 B θ . 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 B θ 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 A θ 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 B θ is approximately finite dimensional, first proved by Bratteli and Kishimoto.