A Survey on Morita Equivalence of Quantum Tori

This paper is a survey on the problem of classifying non-commutative tori up to Morita equivalence and will review the necessary background and discuss some results concerning this question (see [28],[29] and [22]). The concept of Morita equivalence was first introduced in operator algebras by M.Rieffel in the 1970’s, in connection with the problem of characterizing representations of locally compact groups induced by representations of (closed) subgroups (see [20], [14], [15] and also [21]). It provided, in particular, a new proof of Mackey’s imprimitivity theorem, in terms of group C∗algebras. Since then, Morita equivalence has become a very important and useful tool in the theory of C∗algebras (see [23], [19] and the references therein). Recently, the concept of Morita equivalence has been proven to be relevant also in physics, in relation to applications of non-commutative geometry to M(atrix)-theory. In fact, it was shown in [6] that one can consider compactifications of M(atrix)-theory on non-commutative tori and in [29], it was proven that compactifications on (completely) Morita equivalent noncommutative tori are in some sense physically equivalent. The present paper is organized as follows: the first section gives a brief introduction to Morita theory for unital rings and describes how one can adapt the main ideas to the category of C∗ algebras; the second section discusses smooth and topological non-commutative tori first, through a purely C∗algebraic point of view and then using (strict) deformation quantization; finally, the last section will discuss the problem of classifying noncommutative tori up to Morita equivalence. In what follows, the terms non-commutative tori and quantum tori will be used interchangeably. I would like to thank M. Anshelevich, D. Markiewicz and Professors A. Weinstein and M. Rieffel for helpful discussions and comments.