Generators of the $ K_{0} $-group of the non-commutative torus $ A_{\theta} $ with $ \theta \in \mathbb{Q} $ (i.e. rational rotation algebra)

Question

I am studying the non-commutative torus $ A_{\theta} $.

When $ \theta $ is irrational, $ {K_{0}}(A_{\theta}) $ is generated by $ [1] $ and $ [p_{\theta}] $.

(Note: $ p_{\theta} $ is a projection in $ A_{\theta} $, called the Powers-Rieffel projection, and satisfies $ \tau(p_{\theta}) = \theta $, where $ \tau $ is the unique tracial state on $ A_{\theta} $.)

When $ \theta $ is rational, we have the same $ K $-theory as $ A_{\theta} $ is Morita equivalent to $ C(\mathbb{T}^{2}) $. However, I would like to know an actual generating set for $ {K_{0}}(A_{\theta}) $. Could anyone give an example or provide a reference?

I thank you all in advance.

Answer 1

You should be able to find the construction in this paper by Rieffel "The cancellation theorem for projective modules over irrational rotation C∗-algebras, Proc. London Math. Soc. 47(1983), 285–302"

For more general (higher dimensional) rotation algebras, look in Rieffel's paper "Projective modules over higher dimensional non-commutative tori." Just Google it and you should be able to find a copy.

The same technique works in both the rational and irrational cases and gives an explicit description of projective modules over rotation algebras in terms of translation and modulation operators. For the case of the noncommutative 2-torus you mentioned above, the operators are $f \rightarrow f(x-1)$ and $f \rightarrow e^{2 \pi i \theta x}f$ acting on $L^2(\mathbf{R}).$ This module represents the nontrivial generator in $K_0$ and the projection onto this module is equivalent to the Rieffel projection when $\theta$ is irrational.

For an explicit description of the corresponding projections using frame theoretic techniques, you can look in Franz Luef's paper "Projections in noncommutative tori and Gabor frames" http://arxiv.org/abs/1003.3719.

Answer 2

Rieffel's explicit construction of a projection $p_\theta \in A_\theta$ with $\tau(p_\theta) = \theta$ works for rational $\theta$ as well (assuming $0 < \theta < 1$). Further, the generators of $K_0(A_\theta)$ are just $[1]$ and $[p_\theta]$. It is exactly the same as the irrational case. Viewing $A_\theta$ as a crossed product $C(\mathbb{T}) \rtimes \mathbb{Z}$, one obtains a Pimsner-Voiculescu six-term exact sequence of $K$-theory groups. The sequence shows immediately that $K_0(A_\theta) \cong \mathbb{Z} \oplus \mathbb{Z}$, and one can do explicit computations with the boundary maps in the sequence to verify that $[p_\theta]$ is the second generator.