28 2. Nuclear and Exact C* -Algebras
Exercise 2.1.7 (Dependence on range). If B: A-+ Bis nuclear and CC B
is a C*-subalgebra with the properties that (a) B(A) C C and (b) there
exists a conditional expectation ~: B-+ C, then e: A-+ C is also nuclear.
Exercise 2.1.8 (Dependence on range). If 8: A-+ Bis nuclear and CC B
is a C*-subalgebra with the properties that (a) B(A) C C and (b) there
exist a sequence of c.c.p. maps ~n: B -+ C such that ~nlc -+ idc in the
point-norm topology, then e: A -+ C is also nuclear ..
Definition 2.1.5. If A is unital, a~ extension 0 -+ J -+ A ~ A/ J -+ 0 is
called locally split if for each finite-dimensional operator system E C A/ J
there exists a u.c.p. map O': E-+ A such that 7r o O' = idE.
Exercise 2.1.9 (Quotients). Let B: A-+ B be a unital nuclear map such
that BJJ = 0 for some ideal J <1 A. First show that e descends to a u.c.p.
map iJ: A/ J-+ B. Next, show that if 0 -+ J-+ A ~ A/ J-+ O is locally
split, then iJ is nuclear. (Hint: You will need Arveson's Extension Theorem
again.)
2.2. Nonunital technicalitiesMany arguments are more transparent in the presence of units; hence we
will reduce to this case whenever possible. The purpose of this section is to
collect a number of technical facts needed for this reduction. The results are
important and anyone wishing to work in this field should read the proofs
at some point in his or her life, but we don't recommend spending too much
time here.
The first issue is the unitization of a c.c.p. map.
Proposition 2.2.1. Assume A is nonunital, B is unital and <p: A-+ B is
a c.c.p. map. Then <p extends to a u.c.p. map rjJ: A-+ B by the formula
rjJ(a + ,\l.A) = <p(a) + ,\lB,
where A denotes the unitization of A.Proof. "All" we have to prove is that rjJ is also completely positive. To do
this, we first consider the double adjoint map <p : A -+ B. Identifying
double duals with enveloping von Neumann algebras, one checks that <p
maps positive operators to positive operators. Since Mn(C) ~ (Mn(C))
for any C*-algebra C, it follows that <p is also completely positive.^2 Iden-
tifying A with A+ <ClA C A, we have thus extended <p to a c.p. map
A.-+ B. Of course, if we get lucky and <p(lA) happ.ens to be the unit
(^2) There are a number of things to check here, due to the identifications involved, but it all
works.