C∗-algebras associated with irrational rotations

Abstract

For any irrational number a let A a be the transformation group C*-algebra for the action of the integers on the circle by powers of the rotation by angle 2πa.It is known that A a is simple and has a unique normalized trace, τ.We show that for every β in (Z + Za) Π [0,1] there is a projection p in A a 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 A a then τ{p) must be in the above set, one can immediately show that, except for some obvious redundancies, the A a are not isomorphic for different a.Moreover, we show that A a and A β are strongly Morita equivalent exactly if a and β are in the same orbit under the action of GL (2, Z) on irrational numbers.