For any irrational number α
let Aα be the transformation group C∗-algebra for the action of the integers on the
circle by powers of the rotation by angle 2πα. It is known that Aα is simple and has a
unique normalized trace, τ. We show that for every β in (Z+ Zα) ∩ [0,1] there is a
projection p in Aα with τ(p) = β. When this fact is combined with the very
recent result of Pimsner and Voiculescu that if p is any projection in Aα
then τ(p) must be in the above set, one can immediately show that, except
for some obvious redundancies, the Aα are not isomorphic for different α.
Moreover, we show that Aα and Aβ are strongly Morita equivalent exactly if
α and β are in the same orbit under the action of GL(2,Z) on irrational
numbers.
