The Bicategory of m-regular Involutive Quantales

Recently the theory of Morita equivalence for involutive quantales and the notion of the interior tensor products of Hilbert modules over involutive quantales evolved considerably (see e.g. Paseka, 2002 and Paseka, 2001). The present paper is an attempt to put a part of this theory in a broader context of the bicategory of m-regular involutive quantales. For facts concerning quantales in general we refer to Rosenthal (1990), for definitions and motivation concerning involutive quantales we recommend Mulvey and Pelletier (1992). A version of Morita’s theory appropriate for C*-algebras was developed by Rieffel (see Rieffel, 1974 and Raeburn and Williams, 1998) and Blecher (see Blecher, 2001). It is also well known, for example (see Landsman, 2001a and Landsman, 2001b), that C*-algebras form a bicategory and two C*-algebras A, B are isomorphic in this bicategory iff they are Morita equivalent. Here we shall present a translation of this result to m-regular involutive quantales. The plan is as follows: in Section 2 we shall review some basic facts concerning Hilbert modules and modules over involutive quantales in general and then we recall the notion of a bicategory. In Section 3 we subsequently explain the bicategory of m-regular involutive quantales in some detail, including the pertinent Morita theory.