Vol. 93, No. 2, 1981

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Cāˆ—-algebras associated with irrational rotations

Marc Aristide Rieffel

Vol. 93 (1981), No. 2, 415ā€“429

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.

Mathematical Subject Classification 2000
Primary: 46L55
Secondary: 46M20, 58F15
Received: 2 January 1980
Published: 1 April 1981
Marc Aristide Rieffel