1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

13.4. Nonsemisplit extensions 387

CJ into E9~=l Mn(<C), and the c.c.p. map W 0 idn induces an isometric

*-homomorphism 7r:

Proof. Let'!/; be as in Lemma 13.4.4 and 'l/;n: CF--> CF be the c.c.p. map

such that'!/; = ('l/;n)· Choose a dense sequence {Yj}~ 1 in CF 0 D. Since

CF is residually finite-dimensional by Theorem 7.4.1, for each n, there is a

*-homomorphism O'n: CF--> Mk(n) (<C) such that

for every j = 1, ... , n. We may assume that {k(n)} is increasing and define

w: CF--> IJ~=lMk(n)(<C) by w = (crno'l/;n)n· It is not hard to see that w is

multiplicative modulo E9~=l Mk(n)(<C) and W maps CJ into E9~=l Mk(n)(<C).

We regard IJ~=l Mk(n) (<C) as a "lacunary" subalgebra in f1~ 1 Mk(<C). Now

the *-homomorphism 7r in the diagram is well-defined and, for every j, we

have

n->oo

= limsup ll('l/;n 0 idn)(Yj)ll

n->oo

by Lemma 13.4.4. This implies that 7r is isometric. D

We are now in a position to prove Theorem 13.4.1.

Proof of Theorem 13.4.1. Let D = F and take w: CF --> IJ~=l Mn(<C)
as in Lemma 13.4.5. We embed IJ~=l Mn(<C) into JE(f^2 ) via a unitary iden-
tification E9~=l .e~ ~ 1'.^2. Let B C JE(f^2 ) be the (nonunital) C-algebra
generated by w(CF) and IK = IK(f^2 ). It follows that B/IK is -isomorphic
to CF/CJ = CA. Since A is QWEP, the left hand side of the bottom line
of the diagram in Lemma 13.4.5 is canonically *-isomorphic to CA 0max F
(Exercise 13.3.5). It follows that the bottom row of the following commuting
diagram is exact (the top row is also exact, by universality of the maximal