2.2. Nonunital technicalities 29of B, then we are done. Though that won't be the case in general, we do
have 0 S cp(lA) S llc,olllB S lB.
Now we are in a position to check that rp is completely positive. Let 0::::;
[aij+AijlA] E Mn(A) be given. Then 0n([aij+AijlA]) = [cp(aij)+AijlB]
and it suffices to show that 0n([aij+AijlA]) 2 (cp)n([aij+AijlA ]), since
we already observed that cp is completely positive. But,
0n([aij + AijlA]) - (cp)n([aij + AijlA]) = [Aij(lB - cp(lA ))]
= [AijlB]diag(lB - cp(lA )),
where diag(lB -cp(lA )) E Mn(B) is the diagonal matrix with constant
entries lB - cp(lA) down the diagonal. Since Mn(CC) is a quotient of
Mn(A) and 0 S [aij + AijlA] E Mn(A), it follows that [AijlB] 2 0. Note
also that this scalar matrix commutes with diag(lB - cp(lA )) 2 0. Since
the product of commuting positive operators is still positive, it follows that
[AijlB]diag(lB - cp(lA )) 2 0 as desired. D
Remark 2.2.2. The norm of rp may be larger than that of cp (if and only
if llc,oll < 1), but the fact that it is unital usually outweighs this deficiency.
Though we won't need it, note that the proof above still works if one requires
that l_A 1--t llc,olllB, and this produces a map with the same norm as that of
cp.
We will also need the following extension result.
Lemma 2.2.3. If A is unital, Bis unital and cp: A --t Bis c.c.p., then one
can extend cp to a u. c. p. map rp : A EEl C --t B by
rp(a EEl .:\) = cp(a) + .:\(lB - cp(lA)).Proof. First note that 0 ::::; lB - cp(lA) since cp is positive and contractive.
Hence A 1--t .:\(lB - cp(lA)) defines a c.p. map CC --t B. It is a general (and
easily verified) fact that the sum of two c.p. maps is again c.p.; thus the
proof is complete. D
Proposition 2.2.4. Assume e: A --t B is nuclear. If A is nonunital and B
is unital, then the u.c.p. extension given by Proposition 2.2.1 is also nuclear.
If both A and B are nonunital, then the unique unit.al extension iJ: A --t B
is also nuclear.Proof. Let C,On: A --t Mk(n) (CC) and 'I/Jn: Mk(n) (CC) --t B be c.c.p. maps
converging in the point-norm topology toe. In the case that A is nonunital
but B has a unit we first extend the maps 'Pn to u.c.p. maps
lpn: A --t Mk(n) (CC) EEl CC
by regarding Mk(n)(CC) as a subalgebra of Mk(n)(CC) EElCC and applying Propo-
sition 2.2.1. We then use the previous lemma to extend the 'l/Jn's to u.c.p.