<i>C</i>^∗-algebras associated with irrational rotations

TLDR

The transformation group C*-algebra \(A_a\), defined by an irrational rotation of the circle, is simple and possesses a unique normalized trace \(\tau\). We prove that every trace value \(\beta\) in \((\mathbb{Z}+\mathbb{Z}a)\Pi[0,1]\) is realized by a projection in \(A_a\), that distinct irrational parameters \(a\) yield non‑isomorphic algebras, and that \(A_a\) and \(A_\beta\) are strongly Morita equivalent precisely when \(a\) and \(\beta\) lie in the same \(\mathrm{GL}(2,\mathbb{Z})\) orbit. No further metadata is provided.

Abstract

For any irrational number a let Aa 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 Aa 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 Aa 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 Aa then τ{p) must be in the above set, one can immediately show that, except for some obvious redundancies, the Aa are not isomorphic for different a. Moreover, we show that Aa and Aβ are strongly Morita equivalent exactly if a and β are in the same orbit under the action of GL (2, Z) on irrational numbers. 0* Introduction* Let a be an irrational number, and let S