1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

30 2. Nuclear and Exact C* -Algebras

maps ,(/;n : Mk( n) ( <C) EB <C --+ B and one checks that { ,(/;n o <Pn} converges in

the point-norm topology to e.^3

In the case that both A and Bare nonunital we get u.c.p. maps <Pn: A--+

Mk(n)(<C)EB<Cjust as in the previous paragraph. Regarding Bas a subalgebra

of Band the 1/Jn's as taking values in B, we again invoke the previous lemma

to get u.c.p. maps ,(/;n: Mk(n)(<C) EB <C--+ B. Another standard computation

completes the proof. D

The previous result will usually allow us to assume that our algebras are

unital. In most cases, we can further reduce to the case of unital maps.

Lemma 2.2.5. If 0: A --+ Mn(<C) is c.p. and A is unital, then there exists

a u. c. p. map <p : A --+ Mn ( <C) such that for all a E A we have

1 1

0( a) = 0(1A) 2 <p( a )rp(lA) 2.

Proof. In the case that 0(1A) is an invertible matrix this result is trivial

as one simply defines

1 1

cp(a) = 0(1A)-2r:p(a)rp(lA)-2

for all a E A. The general case is more technical but similarly simple.

If P denotes the projection onto the kernel of 0(1A) and PJ_ = 1-Pis

the orthogonal complement, then

rp(a) = PJ_rp(a) = rp(a)PJ_

for all a E A. Evidently it suffices to see this in the case that 0 :::; a :::; lA

and then it is a consequence of the fact that 0 :::; rp(a) :::; rp(lA) (since this

implies the kernel of 0(1A) is contained in the kernel of rp(a)).

Applying the trick from the first part of the proof, we can find a u.c.p.

map <p1: A--+ PJ_Mn(<C)PJ_ as in the statement of the lemma. To complete

the proof, we just take any state 'f}: A --+ <C and define a u.c.p. map <p: A--+

Mn(<C) by <p(a) = <p1(a) EB rJ(a)P. D

Proposition 2.2.6. If(): A --+ B is a unital nuclear map, then there exist

u.c.p. maps <fn: A--+ Mk(n)(<C) and 1/Jn: Mk(n)(<C) --+ B such that 1/Jn o <{Jn --+

() in the point--norm topology.

Proof. Let 0n: A--+ Mk(n)(<C) and ,(/;n: Mk(n)(<C)--+ B be c.c.p. maps whose
compositions converge in the point-norm topology to (). By the previous
lemma we can find u.c.p. maps <{Jn: A-+ Mk(n)(<C) such that
1 1
<Pn(a) = 0n(lA)2<pn(a)<fn(lA)2

for all a E A. These will be the first replacements we need.

(^3) 0£ course, one now applies Exercise 2.1.2 to complete the proof.