1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

11.2. NF and strong NF algebras 321

Proof. We will choose 6 > O later. Let A = EBk= 1 JE(e~(k)) c JE(ea) with

EB e~(k) = ea, let B = EBf=1 IIB(e~(l)) c JE(e;) with EB e~(l) = e;, and let

cp = EB cpz where cpz: A ---t JE(e~(l)). (On a first reading, it will help to

assume s = t = 1.) By Stinespring's Theorem, there are Hilbert spaces Hz

and isometries Vi : e;,(z) ---t e~ Q9 Hz such that

<pz (x) = Vz* (x Q9 l)Vi

for x E A. Let {(}k)}t~k{ be the standard orthonormal basis for e~(k) and

consider the unit vector

d(k)

c 1 ~ r(k) r(k) fJ2 fJ2

c,,k = VdJ}25 f;;t_ i.,,i Q9 i.,,i E .r:,d(k) Q9 .r:,d(k) ·

If {e~~} are the standard matrix units of JE(e~(k)), then

d(k)

Ek= d(k)-^1 L e~~ Q9 e~~ E JE(e~(k) Q9 e~(k)) c A Q9 A

i,j=l

is the rank-one orthogonal projection onto C~k· Since

1- 6 < \(idA Q9 cp)(Ek)ll,

one can find a partition {Kz}f= 1 of the k's with the property that k E Kz
implies ll(idA Q9 cpz)(Ek)ll > 1-6. Let Pz = ViVz*. Then, for every k E Kz,
we have

ll(le2Q9Pz)(Ek@1Hz)lld^2 = ll(idA Q9 cpz)(Ek)ll > 1-6.
Hence, there is a unit vector 'flk E Hz such that 11(1 Q9 Pz)(~k Q9'f/k)11^2 > 1-6.
Let Qk E JE(ea Q9 Hz) be the orthogonal projection onto e~(k) Q9 C'flki clearly
the Qk 's are mutually orthogonal. Letting Rz = :Z:::::kEKz Qk, we have

11(1-Pz)Rzll^2 ~ 11(1-Pz)Rzll~,Tr

d(k)

= LL 11(1-Pz)((i(k) Q9 'f/k)ll^2

kEKz i=l

d(k)

= L 11(1 Q9 (1-Pz))(L(fk) Q9 (fk) Q9 'f7k)ll2

kEKz i=l

= L d(k)(l - ll(l Q9 Pz)(fa Q9 'f7k)ll^2 ) < d6.

kEKz

It follows that if we choose 6 > 0 small enough, then by Lemma 7.2.2 we
may find a unitary operator Ui on ea@ Hz such that l\Uz - lll ~ e/2 and
Uz Rz Uz ~ Pz. Defining 'ljJ = EB 7/Jz, where 7/Jz ( x) = Vz Uz* ( x Q9 1) Uz Vi, we have