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.


2 Answers 2


I guess you want the generators for the $K_0$ part (actually the generators for the $K_1$ part are interesting too, as the 4-th generator (besides the obvious 3 unitaries which generate the algebra) is somewhat non-trivial to construct.

The key idea to write down the corresponding modules is to notice that the bimodules are given as $\mathcal{S}(\mathbb{R}^p \times \mathbb{Z}^q\times F)$, but there exist different representations of the algebra on them, and distinct representations lead to distinct classes in the K-theory.

I am going describe how this works for the 2-torus, as all the interesting things happen already there, and the n-torus case can be extrapolated easily from there. The modules here are either the trivial ones ($\simeq \mathcal{S}(\mathbb{Z}^2)$ as modules), or $\mathcal{S}(\mathbb{R})$.

If $\theta$ is irrational, we have $K_0(\mathbb{T}^2_\theta) = \mathbb{Z}\oplus\theta \mathbb{Z}$, and the algebra $\mathbb{T}^2_\theta$ has cancellation (i.e. if $V\oplus W \simeq V'\oplus W$, then $V \simeq V'$ for projective modules $V,V',W$), hence we have for each $(\mathbb{Z} + \theta \mathbb{Z})^+$ a projective module that represents the class.

The projections corresponding to these classes are the Powers-Rieffel projections, as described in many places. What seems to be less well-known is how to find the corresponding projective module. This was actually written down by Rieffel in "$C^*$-algebras associated with irrational rotations" and "The cancellation theorem for projective modules over irrational rotation $C^*$-algebras".

Rieffel observed that given a strong Morita equivalence bimodule between two $C^*$-algebras $A$ and $B$, with $A$ and $B$-valued inner products $\langle , \rangle_{A}$ and $\langle , \rangle_{B}$, the element $\langle x,x\rangle_A\in A$ is a projection if and only if $ \langle x,x\rangle_B x = x$. The modules $\mathcal{S}(\mathbb{R})$ are the equivalence bimodules used for the noncommutative torus.

For all integers $p,q$, $q\neq 0$, the smooth subalgebra of the $C^*$-algebra of $\mathbb{T}^2_\theta$ acts on $\mathcal{S}(\mathbb{R})$ on the right, with action given by $f(t)\triangleleft U_1 = f(t-pq+q\theta)$ and $f(t)\triangleleft U_2 = e^{2\pi i t/q} f(t)$, for the generators $U_1$ and $U_2$ of the noncommutative $2$-torus. If $a,b\in\mathbb{Z}$ are such that $aq-pb=\pm 1$, there is a left action of the algebra $\mathbb{T}^2_{(a\theta +b)/(p+q\theta)}$, which commutes with this right action.

A Powers-Rieffel projection can be constructed from such a module, for example, using a function $f$ which is strictly positive on the interval $[0,\theta]$ and supported on some interval with length less than $1$, and then appropriately normalizing this function, so that it's inner product becomes $1_B$.

The trace of $\langle f,f\rangle_A$ then becomes(as described in Theorem 1.4 of the cancellation theorem paper) $|q \theta + p|$. If we take an extra finite group $\mathbb{Z}/d\mathbb{Z}$ in the bimodule, the trace becomes $d|q \theta + p|$. Hence the modules $\mathcal{S}(\mathbb{R}\times F)$, with the appropriate action, give you the appropriate classes of $K_0$.

For higher-dimensions, we have more Powers-Rieffel projections, coming from choosing different actions on the various $\mathcal{S}(\mathbb{R}^p\times \mathbb{Z}^q \times F)$.

EDIT I should add that unfortunately, I was too optimistic about the actual computation of the actual projections. If $p=1, q=n-2$, the computation can be done (and leads to projections with trace $\theta_{ij}$), however, I am unable to compute explicit projections coming from modules with $p>1$, and various discussions with other people in the field lead me to believe this is actually an open problem.

  • $\begingroup$ Do you know how to construct Powers-Rieffel projections without going into modules for 3-tori? $\endgroup$
    – SiOn
    Apr 22, 2014 at 17:54
  • $\begingroup$ I'll try to write that down sometime later this week. I'm not aware of any explicit references, but it shouldn't be too hard, given this construction. I now see that there are some copy/paste errors, with sentences which are cut off in weird places, I'm sorry for that. $\endgroup$ Apr 22, 2014 at 19:59
  • $\begingroup$ Could you be a little more explicit about how does the action looks when we add this finite group? $\endgroup$ Aug 18, 2017 at 0:26

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.


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