Counterexamples
K-Homology and
K-Theory
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Chapter 17
In our final chapter, we present an application of finite-dimensional approx-
imation properties to a natural problem in analytic K-homology. Shortly
after the pioneering work of Brown, Douglas and Fillmore, Anderson con-
structed an example of a C* -algebra A for which the Ext semigroup is not a
group ([7]). Since then a number of other counterexamples have been given.^1
Here we will present Simon Wassermann's examples [195], using residually
finite property (T) groups.
In Section 17.2 we prove a structure theorem for C*(r), where r is a
property (T) group. Using this result, it is easy to show that some natural
extensions (actually, the same ones used by Voiculescu in the last chapter)
are not invertible, and thus Ext need not be a group.17.1. BDF preliminaries
The goal ofBDF (Brown-Douglas-Fillmore) theory is to classify extensions of
a fixed algebra A by the compact operators K The original motivation was
a natural problem in single operator theory-the classification of essentially
normal operators - and the solution required importing topological ideas
into the C* -world. This section contains the necessary definitions and a few
lrn fact, Sections 13.4 and 13.5 describe counterexamples of Kirchberg and of Haagerup and
Thorbj!ilrnsen, respectively.
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