Chapter 4
Constructions
There are numerous ways of creating new C* -algebras out of old ones. Our
goal in this chapter is not to study the constructions per se, but rather to
see how nuclearity and exactness behave. As such, the reader unfamiliar
with crossed products, free products or Cuntz-Pimsner algebras will likely
find our treatment lacking in several areas.
In recent years, there have been attempts to unify as many construc-
tions as possible into a single point of view - i.e., to find a general procedure
which contains several well-known constructions as special cases. Elegant
and efficient as this generality may be, we take a different approach, mean-
dering from the particular to the very general; this leads to redundancy, but
we feel it benefits the novice. This also allows us to expose some techniques
that may be useful in other contexts. For example, we construct explicit ap-
proximating maps on crossed products by Z while using a different approach
- fixed point subalgebras of compact group actions -to prove nuclearity of
graph algebras.
4.1. Crossed products
One of the most important constructions in operator algebra theory arises
from noncommutative dynamical systems: the crossed product of a C -
algebra by a group action. Indeed, this idea goes back to von Neumann (in
the context of measurable transformations of a measure space) and has since
given rise to some of the most important modern examples of C -algebras
such as irrational rotation algebras, (stabilized) Cuntz algebras, (stabilized)
graph algebras and others.
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