Extended rotation algebras: adjoining spectral projections to rotation algebras (Q2883844)

This article concerns \(C^*\)-algebras closely related to the well-studied rotation algebras \(A_\theta\), for irrational numbers \(\theta \in (0,1)\). \(A_\theta\) is the universal \(C^*\)-algebra generated by two unitaries \(u\) and \(v\), satisfying the relation \(uv = e^{2\pi i\theta}vu\). This simple algebra has been studied most prominently by \textit{M. A. Rieffel} [Pac. J. Math. 93, 415--429 (1981; Zbl 0499.46039)], who computed its \(K\)-theory, and by \textit{G. A. Elliott} and \textit{D. E. Evans} [Ann. Math. (2) 138, No. 3, 477--501 (1993; Zbl 0847.46034)], who showed that it is an \(A\mathbb T\) algebra, i.e., an inductive limit of algebras of continuous functions from the circle to finite-dimensional algebras. It was shown by \textit{M. Pimsner} and \textit{D. Voiculescu} [J. Oper. Theory 4, 201--210 (1980; Zbl 0525.46031)] that \(A_\theta\) embeds into a simple AF algebra (inductive limit of finite-dimensional algebras) \(C_\theta\), such that \(K_0(C_\theta) \cong K_0(A_\theta)\) (induced by the embedding) and \(K_1(C_\theta)=0\).NEWLINENEWLINEThe authors of this article look at the \(C^*\)-algebras that may arise by adding spectral projections of \(u\) and \(v\) (in some representation of \(A_\theta\) on a Hilbert space). They first allow any prescribed sets to determine the spectral projections that are adjoined. Here they are able to describe a universal \(C^*\)-algebra \(\mathcal B_\theta\) generated by \(A_\theta\) and the prescribed spectral projections, and give an explicit presentation using a countable set of generators and relations. (Note that \(\mathcal B_\theta\) depends on the prescribed set of spectral projections, despite the notation suggesting otherwise.) They show that this universal \(C^*\)-algebra is nuclear, has a unique trace and a unique maximal ideal. This maximal ideal is, in fact, a direct sum of copies of the compact operators. They also show that, when \(\mathcal B_\theta\) is simple, any automorphism of \(\mathcal B_\theta\) is determined by its restriction to \(A_\theta\).NEWLINENEWLINEAttention is focused particularly on the extended rotation algebra \(B_\theta\) given by adjoining simply the spectral projections of \(u\) and \(v\) onto the interval \([0,\theta)\). This \(C^*\)-algebra is simple. Its ordered \(K\)-theory can be described by \(K_0(B_\theta) \cong K_0(A_\theta)\) (induced by the embedding) and \(K_1(B_\theta)=0\). Their crowning result is that \(B_\theta\) is an AF algebra, for generic irrational \(\theta \in (0,1)\) (i.e., for all \(\theta\) in a dense \(G_\delta\) subset of \((0,1)\)). This gives a concrete realization of the AF algebra \(C_\theta\) described above.NEWLINENEWLINEIn order to prove that \(B_\theta\) is AF for generic \(\theta\), the authors build a continuous field of \(C^*\)-algebras (over \((0,1)\)) whose fibres are \(B_\theta\) at irrational \(\theta\), and are dimension drop algebras at rational points. This continuous field is not the most obvious one: borrowing the irrational \(\theta\) construction yields a field that is not continuous, hence a certain modification is required. Using the fact that dimension drop algebras are semiprojective, it follows that \(B_\theta\) is the inductive limit of dimension drop algebras, for generic values of \(\theta\). The classification of dimension drop algebras then yield that, for such \(\theta\), \(B_\theta\) is AF.NEWLINENEWLINEThe main result of this paper begs the question of whether \(B_\theta\) is in fact AF for all irrational \(\theta\). The authors have announced that they have found a positive answer to this question, by making use of recent breakthrough results of \textit{H. Matui} and \textit{Y. Sato} [``Decomposition rank of UHF-absorbing \(C^*\)-algebras'', \url{arXiv:1303.4371}].