The homotopy groups of the unitary groups of non-commutative tori. (English) Zbl 0656.46056

Let \(\theta\) be a real-valued skew bilinear form on \({\mathbb{Z}}^ m\). The form \(\theta\) is not rational if there exists at least one pair (x,y) in \({\mathbb{Z}}^ m\times {\mathbb{Z}}^ m\) such that \(\theta\) (x,y) is not a rational number. Recall that a non-commutative m-torus \(A_{\theta}\) is the universal \(C^*\)-algebra generated by unitary elements \(u_ x\) for \(x\in {\mathbb{Z}}^ m\) such that \[ u_ yu_ x\quad =\quad \exp (\pi i\theta (x,y))u_{x+y} \] for all \(x,y\in {\mathbb{Z}}^ m\). The main result of the article: If the form \(\theta\) is not rational, then \[ \pi_ k(U_ n(A_\theta)) \cong \left\{ \begin{aligned} K_1 (A_\theta) \quad&\text{for \(k\) even}\\ K_0 (A_\theta) \quad&\text{for \(k\) odd} \end{aligned} \right\} \cong \mathbb{Z}^{2^{m-1}} \] for all \(k\geq0\) and \(n\geq1\).
Reviewer: N.L.Vasilevskij


46L80 \(K\)-theory and operator algebras (including cyclic theory)
55Q52 Homotopy groups of special spaces
46L55 Noncommutative dynamical systems
55N15 Topological \(K\)-theory