10.2. Exact C* -algebras 303Crossed products. For amenable groups everything is nice: The crossed
product is nuclear if and only if the original algebra is too (Theorem 4.2.6).
Otherwise, things get complicated. For example, C(X) ><1ar is nuclear if and
only if a is an amenable action (Theorem 4.4.3). For nonabelian algebras
and exact groups, partial results can be formulated (e.g., Theorem 4.3.4),
but they aren't very general.
Though unfairly neglected in this treatise, locally compact amenable
groups also behave well with respect to crossed products: Full and reduced
crossed products always agree, and the crossed product is nuclear whenever
the algebra being acted upon is nuclear (see [5, Theorem 5.3] for a more
general result).
Free products. The full free product (of unital algebras, with amalgama-
tion over units)^1 of two nuclear C -algebras is almost never nuclear ( C (ID' 2) ~
C('JI')*C('JI') is not even exact, by Proposition 3.7.11). Reduced free products
are a little better.
Theorem 10.1.8. The reduced (amalgamated) free product of two nuclear
C*-algebras is always exact (Corollary 4.8.3).
In the pure-state case things work properly.Theorem 10.1.9. Let A and B be unital nuclear C*-algebras with states <p
and 'lj.J, respectively. Assume <p is pure. Then
(A, <p) * (B, 'l/J)
is nuclear (Theorem 4.8. 7).10.2. Exact C*-algebras
Here's the main theorem regarding exactness:Theorem 10.2.1. For a C*-algebra A, the following are equivalent:
(1) there exists a faithful *-homomorphism 7r: A -----t IIB(H) which is nu-
clear (Definitions 2.1.1 and 2.3.2i see Exercise 2.3.9 as well)i
(2) for every short exact sequence 0 -----t J -----t B -----t B / J -----t 0, the
sequence
0 -----t J 0 A -----t B 0 A -----t (BI J) 0 A -----t 0
is also exact (Theorem 3.9.1 and Exercises 3.9.6 and 3.9. 7).lWhich hasn't been defined, but satisfies the universal property that any pair of unital *-
homomorphisms A-+ lll\(7-l) and B-+ lll\(7-l) extends uniquely to A* B.