Many properties of a C*-algebra (anything K-theoretic, for instance) are unaffected if the algebra is tensored with K. Given a C*-algebra A, we call K ⊗ A the stable algebra of A. A C*-algebra B is called stable if B ≅ K ⊗ B. Of course, the stable algebra of A is always stable since K ≅ K ⊗ K. Two C*-algebras A, B are said to be stably isomorphic if their stable algebras are isomorphic. In this short chapter we shall prove a theorem of Brown, Green and Rieffel which characterises stable isomorphism for σ-unital C*-algebras in terms of a relation called Morita equivalence. We shall not investigate Morita equivalence in any systematic way, but just develop enough of its properties to prove the main theorem. For a fuller treatment, see [Rie 1], [Rie 2], [Bro], [BroGreRie].

Before defining Morita equivalence, we need to investigate some properties of full Hilbert C*-modules. A Hilbert A-module E is said to be full if the ideal 〈E, E〉 is dense in A.

Our first result about full Hilbert C*-modules involves a construction that will be relevant to the notion of Morita equivalence. Let A be a C*-algebra and let E, F be Hilbert A-modules. Under the natural operation of composition of mappings, K(E, F) is a right K(E)-module. (In the same way, k(E, F) is a left K(F)-module. This K(E)-K(F)-bimodule structure is central to the study of Morita equivalence.) Set B = K(E) and G = K{E, F).

This book is designed as a “second course in C*-algebras”. It presupposes a familiarity with the elementary theory of C*-algebras (the GNS construction, the functional calculus) such as may be found in any of the several excellent texts now available—[Dix 2], [KadRin], [Mur], [Ped], [Tak], for example. The aim, as indicated by the subtitle, is to provide the student with a collection of techniques that have shown themselves to be useful in a variety of contexts in modern C*-algebra theory.

These techniques centre round the quite elementary and natural concept of a Hilbert C*-module. As explained in Chapter 1, this is an object like a Hilbert space except that the inner product is not scalar-valued, but takes its values in a C*-algebra. The first three chapters present the elementary theory of Hilbert C*-modules and their bounded adjointable operators. From Chapter 4 onwards, tensor products figure prominently, and some knowledge of tensor products of C*-algebras (summarised at the beginning of Chapter 4) is needed.

Hilbert C*-modules have had three main areas of applications:

• the work of Rieffel and others on induced representations and Morita equivalence ([BroGreRie], [Rie 1], [Rie 2]);

• the work of Kasparov and others on KK-theory ([BaaJul], [Kas 1], [Kas 2]);

• the work of Woronowicz and others on C*-algebraic quantum group theory ([BaaSka], [Wor 5]).

There is not very much information on any of these topics in this book, since the aim is to develop a toolkit rather than to demonstrate the tools in use.