In this chapter, we will develop the Brown-Douglas-Fillmore (BDF) theory of extensions, and the generalization due to Kasparov.

Extension theory is important in many contexts, since it describes how more complicated C*-algebras can be constructed out of simpler “building blocks”. Some of the most important applications of extension theory are:

• (1) Structure of type I C*-algebras, group C*-algebras, and crossed products.

• (2) Classification of essentially normal operators.

• (3) Index theory for elliptic pseudodifferential operators.

• (4) Using associated homological invariants to distinguish between C*-algebras (often simple C*-algebras).

These applications will be discussed in more detail later.

We will not follow the historical development of the theory very closely; in fact, much of what we do will be in reverse historical order. We have chosen to do things this way since much of the general theory discussed in Section 15 must be done in essentially the same way anyway for BDF theory, and it is no more difficult to do things in full generality.

We will not go into all the ramifications of BDF theory; the interested reader may consult [Douglas 1980] for a more complete treatment. [Rosenberg 1982a] is also recommended as a good overall source, including Kasparov's theory. Other references include [Brown 1976; Baum and Douglas 1982b; Valette 1982].