1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

258 7. Quasidiagonal C* -Algebras

for all a, b E J. It follows that 1/Jn is almost multiplicative on !f?n(J). More

precisely,

is bounded above by

111/Jn(Son(a)rpn(b)) -1/Jn(!f?n(ab))ll + 111/Jn(!f?n(ab)) - abll

+ llab-1/Jn(!f?n(a))'l/Jn(!f?n(b))ll

which, in turn, is bounded above by 4c: whenever a, b E J. To ease notation,

we now fix n (large enough) and just write rp: A-+ Mk(C) and 1/J: Mk(q -+

IIB(H) instead of !f?n and 1/Jn·

Let O": Mk(q -+ IIB(JC) be the Stinespring dilation of 1/J and V: 1i-+ JC

be the isometry such that

1/J(x) = V*O"(x)V

for all x E Mk(C). Since we have arranged that 1/J is 4c:-multiplicative on

rp(J), it follows that the Stinespring projection Q = VV* almost commutes

with O"( rp(J)). More precisely, we have

llQO"(rp(a))Q1-ll^2 = 111/J(rp(a)ip(a)*) -1/J(rp(a))'l/J(rp(a))*ll ~ 4c:

and similarly llQ1-0"(rp(a))Q/1^2 ~ 4c:, for all a E J. It follows that

llO"(rp(a)) - (QO"(rp(a))Q EB Q1-0"(rp(a))Q1-) JI ~ 2y!E

for all a E J. But

QO"(rp(a))Q = V'ljJ(rp(a))V*

and hence

llO"(ip(a)) - (VaV* EB Q1-0"(i,O(a))Q1-) II ~ 2y!E + c:

for all a E J. We complete the proof by defining 7r: A -+ IIB(JC) by 7r(a) =

VaV*, <I>: A-+ IIB(JC) by <I>(a) = Q1-0"(i,O(a))Q1-and!= A-+ O"(Mn(C)) C

IIB(JC) by 1(a) = O"(rp(a)). 0

Remark 7.5.6. Note that if we have [Pn, a] = 0 for all a E A (i.e., A is

RFD and the inclusion A c IIB(H) is block diagonal), then the u.c.p. map/

is a *-homomorphism.

Let's take a moment and understand what the previous lemma really
says: In the presence of exactness and quasidiagonality we can dilate a
representation to make it close to something finite-dimensional. This may
remind you of Arveson's proof of Voiculescu's Theorem (cf. [11]) where one
of the major steps was dilating something to be block diagonal. We now
recycle that idea to push our approximation results further.