TITLE: Projective modules over noncommutative tori?
QUESTION [8 upvotes]: It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class functions on $M=\mathbb{R}^p \times \mathbb{Z}^q \times F$, where $F$ is some finite Abelian group and $2p+q=n$.
My question is: if I want to explicitly construct basis of K-theory for simple noncommutative tori ($\mathcal{A}$) of dimension 3, then what would be my choice of $M$'s? There are four modules to be constructed and one is of course trivial module.

REPLY [3 votes]: This is really just an add on to Jan's answer, but explicit projections for these modules can be constructed using Gabor frames. See for example "Projections in noncommutative tori and Gabor frames" http://arxiv.org/abs/1003.3719 by Luef. 
This method works for even dimensional noncommutative tori when p is equal to half the dimension. However, I wouldn't be surprised if the techniques from frame theory could be adapted to construct projections for the other cases, and it would be interesting to see that worked out even just for 3-tori.