1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

122 4. Constructions

where

and so on. Since Aj Ai = 0 whenever i # j, a straightforward calculation
completes the proof. D

Lemma 4.2.2. If A is a I'-C*-algebra and F c r is a finite set, then for
each set { ap}pEF CA, the element

L ap(a;aq)Apq-1 E Cc(r, A)

p,qEF

is positive as an element in A ><1 a r (or A ><1 a,r r).

Proof. The element in question is equal to CI:pEF apAp-1 )* (I:pEF apAp-1).
D

Here are the approximating maps we're after.

Lemma 4.2.3. If A is a I'-C* -algebra and F c r is a finite set, then there
exist c.c.p. maps i.p: A><la,rI'-+ A@Mp(C) and 'I/;: A@Mp(C)-+ Cc(I', A) C
A ><1 a,r I' such that for all a E A and s E r we have

IFnsFI

'I/; o i.p(a>.8 ) = IFI aAs.

Proof. In the proof of Proposition 4.1.5 we saw that there is a c.c.p. map
l.(J: A ><la,r r-+ A® Mp(C) such that

i.p(a>.s) = L a;^1 (a) ® ep,;-1P.

pEFnsF

It suffices to prove that 'I/;: A® Mp(C) -+ Cc(I', A) c A ><la,r r defined by

1

'l/;(a ® ep,q) = IFfo:p(a)>.pq-1

is a c.c.p. map, as a simple calculation confirms the asserted formula.

In fact, it suffices to prove that 'I/; is positive since Exercise 4.1.3 provides
a natural commutative diagram

Mn(C) ®(A® Mp(C)) (Mn(C) ®A) ® Mp(C)

l l

Mn(C) ®(A ><la,r I') (Mn(C) ®A) ><IT@a,r r.