# About the K-theory of rational rotation algebras

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.

• 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!
– Ruy
Sep 24, 2020 at 4:17