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

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.