A model theoretic Rieffel's theorem of quantum 2-torus

We defined a notion of quantum 2-torus $T_{\theta }$ in “Masanori Itai and Boris Zilber, Notes on a model theory of quantum 2-torus $T_q^2$ for generic $q$, arXiv:1503.06045v1 [mathLO]” and studied its model theoretic property. In this note we associate quantum 2-tori $T_{\theta }$ with the structure over ${\mathbb C}_\theta = ({\mathbb C}, +, \cdot, y = x^\theta),$where$\theta \in {\mathbb R} \setminus {\mathbb Q}$, and introduce the notion of geometric isomorphisms between such quantum 2-tori. We show that this notion is closely connected with the fundamental notion of Morita equivalence of non-commutative geometry. Namely, we prove that the quantum 2-tori $T_{{\theta }_1}$ and $T_{{\theta }_2}$ are Morita equivalent if and only if ${\theta }_2={\displaystyle \frac{a{\theta }_1+b}{c{\theta }_1+d}}$ for some $ \left( \begin{array}{cc} a & b \ c & d \end{array} \right) \in {\rm GL}_2({\mathbb Z})$with$|ad - bc| = 1$. This is our version of Rieffel’s Theorem in “M. A. Rieffel and A. Schwarz, Morita equivalence of multidimensional noncummutative tori, Internat. J. Math. 10, 2 (1999) 289-299” which characterises Morita equivalence of quantum tori in the same terms. The result in essence confirms that the representation $T_{\theta }$ in terms of model-theoretic geometry \cite{IZ} is adequate to its original definition in terms of non-commutative geometry.