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.



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


