Non-commutative spheres. III: Irrational rotations
The paper under review is the third part of a of series papers on non- commutative spheres, see \textit{O. Bratteli}, \textit{G. A. Elliott}, \textit{D. E. Evans}, and \textit{A. Kishimoto} [Part I, Int. J. Math. 2, 139-166 (1991; Zbl 0759.46063); part II, J. Oper. Theory (to appear)]. Let \(A_ \theta\) be the rotation algebra, i.e. the universal \(C^*\)-algebra, generated by two unitaries \(U\) and \(V\), satisfying \(VU=e^{2\pi\sqrt{- 1}\theta}UV\), where \(0\leq\theta<1\). The automorphism \(\sigma\), defined over the generators \(U\) and \(V\) as \[ \sigma(U)=U^{-1}, \qquad \sigma(V)=V^{-1} \] of this algebra is often refered to as the flip \(\sigma\). Define \(B_ \theta=A_ \theta^ \sigma\) as the fixed point algebra and \(C_ \theta=\langle\sigma\rangle \ltimes A_ \theta^ \sigma\), the crossed product. In part I and II, the authors and others described \(B_ \theta\), \(C_ \theta\) in terms of generators and relations and in terms of matrix- valued functions over spheres. The main result of this paper asserts that: \(B_ \theta\) and \(C_ \theta\) are AF-algebras when \(\theta\) is irrational. If \(0<\theta_ 1,\theta_ 2<1\) and \(\theta_ 1\), \(\theta_ 2\) are rational then \(C_{\theta_ 1}\) is isomorphic to \(C_{\theta_ 2}\) if and only if \(\theta_ 1\in \{\theta_ 2,1- \theta_ 2\}\). This contrasts, say the authors, with the rational case where the algebras \(C_{p/q}\) and \(C_{p'/q'}\) (with \(p\), \(q\), and also \(p'\), \(q'\), relatively prime) are isomorphic if and only if \(q=q'\) [see Part II, loc.cit.].
scientific article; zbMATH DE number 96715
scientific article