3.9. Exactness and tensor products 111The final step is an immediate consequence of a basic c.b.-perturbation
fact. Indeed, the following lemma is just a special case of Corollary B.11
since each of the maps cp~^1 1'Pn(E) is already self-adjoint and the dimension
of 'Pn ( E) is fixed.
Lemma 3.9.7. With assumptions and notation as in Proposition 3.9.6, it is
possible to find u.c.p. maps 'I/Jn: 'Pn(E) -+A such that 111/Jn -cp~^1 l'Pn(E) II -+ 0.
The proof of Theorem 3.9.1 in the case that A is unital and separable
is now basically complete. The only thing left to observe is that the u.c.p.
maps 'I/Jn: 'Pn(E) -+ A can be extended to u.c.p. maps Ms(n)(C) -+ JIB(H);
hence the argument sketched in the beginning of this section actually works.
The general case is not too hard to deduce from the separable unital case,
so we consider our work here finished.
Exercises
Exercise 3.9.1. If A is nonunital and ®-exact, then its unitization is also
®-exact (Exercise 3.7.3).
Exercise 3.9.2. Since a nonunital C* -algebra is exact if and only if every
unital, separable subalgebra of its unitization is exact, use the previous
exercise and Exercise 3.7.1 to prove the general case of Theorem 3.9.1.
Exercise 3.9.3. Prove that if lK denotes the compact operators, then
lK ® ffl q -+II (JK ® q
N N
is not surjective. In other words, finite-dimensionality is necessary in Lemma
3.9.4.
If you understand the proof of Theorem 3.9.1, the next three exercises
should be easy.
Exercise 3.9.4. Let AC IIB(H) be exact and E C A be a finite-dimensional
operator system. Show that for every E: > 0 there exists a u.c. p. map
cp: A-+ Mn(<C) and a c.b. map 1/J: Mn(<C)-+ IIB(H) such that 111/Jllcb :S 1 + E:
and 'ljJ o cp(x) = x for all x E E. In other words, if one allows c.b. maps,
then exactness is not an approximate factorization property - it is local
factorization on the nose!
Exercise 3.9.5. Let AC IIB(H) be exact and EC A be a finite-dimensional
operator system. Show that if {vi} C 1i is an orthonormal basis and Pn is the
orthogonal projection onto the span of { v1, ... , Vn}, then for all large n there
exist u.c.p. maps 'I/Jn: PnIIB(H)Pn -+ JIB(H) such that 1/Jn(PnxPn) E A and
llx - 1/Jn(PnXPn)ll :S sllxll for all x EE. Why doesn't this imply nuclearity?
Show that if (and only if) A is nuclear, one can force the entire range of
each 'I/Jn into A.