An Inductive Limit Model for the $K$-Theory of the Generator-Interchanging Antiautomorphism of an Irrational Rotation Algebra

http://dx.doi.org/10.4153/CMB-2003-044-0
Canad. Math. Bull. 46(2003), 441-456

  Published:2003-09-01
 Printed: Sep 2003

Let $A_\theta$ be the universal $C^*$-algebra generated by two unitaries $U$, $V$ satisfying $VU=e^{2\pi i\theta} UV$ and let $\Phi$ be the antiautomorphism of $A_\theta$ interchanging $U$ and $V$. The $K$-theory of $R_\theta=\{a\in A_\theta:\Phi(a)=a^*\}$ is computed. When $\theta$ is irrational, an inductive limit of algebras of the form $M_q(C(\mathbb{T})) \oplus M_{q'} (\mathbb{R}) \oplus M_q(\mathbb{R})$ is constructed which has complexification $A_\theta$ and the same $K$-theory as $R_\theta$.