1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.3. Cones over general exact algebras 269

We say cp is (J,E)-suitable if it is (J,E)-multiplicative and there is a u.c.p.

map 'lj;: Mk(CC) ----+ lB(H) such that

ll'I/;^0 cp(a) - all < E

for all a E J.

Remark 8.3.2. It is important (though trivial) to note that if cp is (J, E)-

suitable and 'lj; is (J, E)-multiplicative, then cp E9 'lj; is (J, E)-suitable. That is,

adding multiplicative maps to suitable maps does not destroy suitability.

It follows easily from Arveson's Extension Theorem that the notion of

(J, E)-suitability does not depend on the choice of representation Ac lB(H).

Here is the stable uniqueness result we need this time.

Lemma 8.3.3. Let J C A C JB(H) be a finite set of unitaries and let E > 0.

If cp: A----+ Mp(CC) is an (J, E)-suitable u.c.p. map and 'lj;: A----+ Mq(CC) is an

(J, E)-multiplicative u.c.p. map, then there exists an integer N E N and an

(J, 5.)€)-multiplicative u.c.p. map p: A----+ MNp-q(<C) such that Ncp (-6,~Vc)

'lj; E9 p.

Proof. The proof uses some ideas and calculations from the proof of Lemma
7.5.5 so we describe how to get p and leave the estimates to the reader.
Let a: Mp(<C)----+ JB(H) be a u.c.p. map with llf-o:ocp(f)ll < E for f E J.
We may assume that 'lj; is defined on all of JB(H) and thus get a u.c.p. map
'lj; o a: Mp(CC) ----+ Mq(CC) which is almost multiplicative on cp(J) (since a is
necessarily close to multiplicative on this set and 'lj; is almost multiplicative
on J). By Stinespring's Theorem, we get a *-homomorphism 7r: Mp(CC)----+
JB(JC) and a projection P E JB(JC) such that 'lj; o a can be identified with
compression by P. In this case JC is a finite-dimensional Hilbert space; hence
there is an integer N such that JB(JC) ~ Mp(CC)@MN(CC) and we may identify
7r: Mp(CC) ----+ JB(JC) with the canonical embedding Mp(CC) ----+ Mp(<C)®MN(<C),
T f----+ T ® 1. Thus 'lj; o a can be identified with the map T f----+ P(T ® l)P.

As we have seen, almost multiplicativity implies P almost commutes

with cp(J) ® 1 C Mp(CC) ® MN(CC) and hence, for every a E J,

Ncp(a) = cp(a) ® 1 ~ P(cp(a) ® l)P E9 PJ_(cp(a) ® l)PJ_

~'I/;( a) E9 PJ_( cp(a) ® l)PJ_.

The desired map p: A----+ MNp-q(CC) is defined by p(a) = PJ_(cp(a) ® l)PJ_.

D

Note that Theorem 7.5.7 implies the existence of (J, E)-admissible maps

whenever A is exact and QD. Hence the stable uniqueness result above