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.