Rieffel induction and strong Morita equivalence in the context of Hilbert modules

The Morita equivalence of m-regular involutive quantales in the context of the theory of Hilbert A-modules is presented. The corresponding fundamental representation theorems are shown. We also prove that two commutative m-regular involutive quantales are Morita equivalent if and only if they are isomorphic. In the paper [5] F. Borceux and E.M. Vitale made a first step in extending the theory of Morita equivalence to quantales. They considered unital quantales and the category of all left modules over these unital quantales which are unital in a natural sense. They proved that two such module categories over unital quantales A and B, say, are equivalent if and only if there exist a unital A−B bimoduleM and a unital B−A bimodule N such that M ⊗B N ≃ A and N ⊗A M ≃ B. The aim of this paper is to extend this theory in the following way: to cover also the case of m-regular (generally non-unital) involutive quantales and Hilbert modules over them. Our motivation to work in this setting comes from the theory of operator algebras, where there is a theory of Morita equivalence for C∗-algebras for the non-unital case (see [4], [8] and [12]). Our presentation is a combination of those in [1], [4] and [5]. This paper is closely related to the papers [9] and [11] where the interested reader can find unexplained terms and notation concerning the subject. For facts concerning quantales and quantale modules in general we refer to [13]. The algebraic background may be found in any account of Morita theory for rings, such as [1] or [3]. The paper is organized as follows. First, we recall the notion of a right Hilbert A-module and related notions. In Section 1 the necessary basic properties of right Hilbert modules are established. Moreover, a categorical 2000 Mathematics Subject Classification. 46M15, 46L05, 18D20, 06F07.