3.9. Exactness and tensor products 105and mimic exactly what we did in the proof of Theorem 3.8.5. It is a good
exercise to check the remaining details.
Exercise
Exercise 3.8.1. Prove that extensions of nuclear C*-algebras are nuclear.
That is, if J <l A is a nuclear ideal and A/ J is also nuclear, then so is A.
(It is possible, but not so easy, to prove this directly from the definition of
nuclearity. However, a tensor product argument, invoking the 5 Lemma, is
very easy.)3.9. Exactness and tensor products
Our only goal in this section is to prove the following deep and difficult
theorem.
Theorem 3.9.1 (Kirchberg). A C*-algebra A is exact if and only if for each
C* -algebra B and ideal J <l B the sequence
0 ~ J 0 A ~ B 0 A ~ (BI J) 0 A ~ 0
is exact - i.e., if and only if we always have a canonical isomorphismB 0 A ~ (BI J) 0 A.
J0A
We already observed the "only if' direction in Proposition 3.7.8; hence
we are left to prove the converse. Throughout this section we say that A is
0-exaci?7 if for each C* -algebra B and ideal J <l B the sequence
0 ~ J 0 A ~ B 0 A ~ (BI J) 0 A ~ 0
is exact; our goal is to show that every 0-exact C* -algebra is exact.
The proof is quite technical and requires meticulous care as there are a
number of identifications that one must keep track of. Before diving into the
details, let us give the main idea so you can keep the big picture in mind.
Suppose that A c Illl(H) is given, Pn E Illl(H) are finite-rank projections
converging to the identity (strong operator topology) and E c A is a finite-
dimensional operator system. We can always define u.c.p. maps cpn: A ~
Ms(n)(C) ~ Pnllll(H)Pn by cpn(a) = PnaPn and, by finite-dimensionality,
the restrictions 'PnlE: E ~ Ms(n) (C) will be injective linear maps for all
sufficiently large n. Hence we can consider the inverse maps cp~^1 l<p,,,(E)
'Pn ( E) ~ E. Now assume a miracle occurs and we know that
llcp;;:^1 1<p,,,(E)llcb ~ 1.27This is just convenient terminology used primarily in this section.