I've been working with Rieffel's "Projective Modules over Higher-dimensional Non-commutative Tori" and I'm struggling with a few basic questions.
I know that when the dimension $d=2$ we have that $T_\theta$ is specified by a single (say irrational) number $\theta$, and then $K_0(T_\theta) = \mathbb{Z} + \theta \mathbb{Z}$. Further, classes $a+\theta b$ in $K_0$ are represented by actual vector bundles if the "rank", or more appropriately the trace, satisfies $a+\theta b > 0$.
Now in general there is a formula (Theorem 3.4) in this paper which gives the trace $$\tau([V]) = |G/D^\perp|$$ if $V$ is the projective module over $T_\theta$ given by the Schwartz space $\mathcal{S}(M)$ with $\mathbb{Z}^d \simeq D \hookrightarrow G = \hat M \times M$ a cocompact embedding of a lattice into a product $\hat M \times M$ of a locally compact abelian group and its dual, and the action of $T_\theta$ is given by restricting the natural action of $\hat M\times M$ on $\mathcal{S}(M)$. The only ingredient left is the normalizaton of the Haar measure on $G$ which would let us calculate $|G/D^\perp|$, which the paper says is given by any normalization on $M$ and then the Plancharel measure on $\hat M$.
Is there a concise description of the order structure on $K_0(T_\theta)$ when $d = 3$ or $d=4$ or higher dimensions as there is when $d=2$?
