On Rieffel's theorem (Model theoretic aspects of the notion of independence and dimension)

Abstract

Let thetain mathbb{R}backslash mathbb{Q}. We defined a notion of quantum 2-torus T_{theta} in [1] and studied its model theoretic properties. In the subsequent paper [2], we introduced the notion of geometric equivalence and also of Morita equivalence between such quantum 2-tori. We showed that this notion is closely connected with the fUndamental notion of Morita equivalence of non-commutative geometry. Namely, we proved that the quantum 2-tori T_{theta_{1}} and T_{theta_{2}} are Morita equivalent if and only if theta_{2}=frac{atheta_{1}+b}{ctheta_{1}+d} for some begin{aray}{l} a b c d end{aray}in GL_{2}(mathbb{Z}). This is our version of Rieffel's Theorem [4] which characterizes Morita equivalence of quantum tori in the same terms. In this note we reconsider the relation between the original version of Rieffel's theorem and our model theoretic version.