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The K-theory of the non-commutative torus or rotation algebra $A_\theta$ was studied in the early 80's by several mathematicians with a special focus on the irrational case. However I have not been able to find a proof for the fact that $$K_0(A_\theta)=\mathbb{Z}[1]\oplus\mathbb{Z}[p_\theta],$$ where $p_\theta$ is a Power-Rieffel projection and $\theta$ is rational.

What I've been able to find in the rational rotation case are the next results:

  • There exist a Power-Rieffel projection $p_\theta\in A_\theta$ and a continuous trace state $\tau_\theta$ on $A_\theta$ such that $\tau_\theta(p_\theta)=\theta$ (Gracia-Bondía, Varilly and Figueroa. Elements of Non-Commutative Geometry).
  • Every trace state on $A_\theta$ induces the same map in $K_0(A_\theta)$ (Elliot. On the K-theory of the C-algebra generated by projective representation of a torsion free discrete abelian group).

Observe that in the irrational case, the original prove given by Pimsner and Voiculescu consisted on proving that the trace state $\tau_\theta$ has a specific range, namely $\mathbb{Z}\oplus \theta\mathbb{Z}$, and then since $\tau(p_\theta)=\theta$ the result was stated. However for the rational case I have not been able to find a similar result.

Any reference will be highly appreciated.

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    $\begingroup$ I believe you will find references and answers to all of your questions in mathoverflow.net/questions/177753/…. To a large extent, the rational and irrational cases are essentially the same with the same proofs (except perhaps for the fact that the trace might not induce an embedding into $R$, since $Z+\theta Z$ is not isomorphic to $Z^2$). Unfortunately an excessive emphasis on the irrational case left lots of holes in the literature! $\endgroup$
    – Ruy
    Sep 24, 2020 at 4:17

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