17. Basic Theory

One of the principal features of Kasparov theory is the great generality of the definitions. In addition to the fact that the theory works for real and “real” C*- algebras as well as for complex ones, and allows a locally compact group action, the definitions are made in such a way that the objects are extremely general. This generality is both an advantage and disadvantage. The advantage is that the objects of KK-theory can be recognized in a variety of quite different situations, leading to broad applicability. Some of the contexts in which KK theory arises are given below, and others in section 24. On the other hand, the generality of the objects creates a certain lack of concreteness which appears to make the theory difficult to understand.

It is difficult to motivate the most general and useful form of the definition from the point of view taken so far in these notes, although some motivation is given in 12.2.4, 15.7.2, and 16.3. The best motivation comes from the theory of pseudodifferential operators (17.1.2(e)). So rather than trying to give any more motivation at first, we will simply plunge in and see later that all the If-groups and Ext-groups considered so far are special cases of KK-groups.

We will begin with the general definitions, and then present some simplifications which can be made in the theory which lead to some concrete descriptions of the objects from various points of view. The two principal points of view we will take are the Fredholm Module Picture, which is essentially Kasparov's original viewpoint, and the Quasihomomorphism Picture due to Cuntz.