1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

326 11. Simple C*-Algebras

To finish the proof one considers the representation

1T = 1T1 EB ... EB ?Tj: A-----+ JB(H1 EB ... EB Hj)

and the projection PE JB(H1 EB··· EB Hj) obtained by adding up P1, ... , Pk

(using the identifications JB(Hi) ~ JB(Hm) for some m :::::; j whenever i > j).

Since 7T 1 , ... , ?Tj are pairwise disjoint, P E ?T(A)". Finally, P commutes

with ?T(A\O), compression by P recaptures 'PIA'P, and this easily implies that

P almost commutes with ~ and approximately preserves the norms after

compression. D

Here is the main technical result. It depends on Smith's lemma (Lemma

B.4), which you may want to review before reading the proof.

Proposition 11.3.8. The unital C* -algebra A is inner QD if and only if

it satisfies the following approximate factorization property: For every finite

set ~ C A, c: > 0 and every u.c.p. map <p: A -----+ Mk(<C), there exist a

projection p in the socle of A** and a u.c.p. map 'ljJ: pA**p-----+ Mk(<C) such

that

(1) ll[a,pJll < c: and

(2) 117/J(pap) - <p(a)ll < E for all a E ~-

Proof. The "if' direction is trivial since one can always find u.c.p. maps
'Pn: A-----+ Mk(n)(<C) such that llall = lim ll'Pn(a)ll for all a EA (just cut by
finite-rank projections in some faithful representation).
To prove the "only if" direction, let's take Pn as in Definition 11.3.1 and
denote by (}n: A -----+ PnApn the corresponding compressions. Since the (}n's
are asymptotically multiplicative and asymptotically isometric, one has
n~~ ll(idMk(IC) 0 (}n)(x)ll = llxllMk(IC)®A
for any fixed k EN and x E Mk(<C) @A. (Why?) Now suppose that a finite-
dimensional operator system EC A, E > 0 and a u.c.p. map <p: A-----+ Mk(<C)
are given. Note that (}nlE is a linear isomorphism for sufficiently large n,
hence we can let e;;-^1 : (}n(E) -----+ E denote the inverse. It follows from the
above equality and Lemma B.4 that
limsup ll'P o 8~^1 llcb = limsup llidMk(IC) 0 (<po 8~^1 )11
n n
:S limsup ll(idMk(IC) 0 (}nlE)-^111
n
:::::; 1.
Hence, we can find n such that ll'P o e;;-^1 llcb < 1 + c/2. We may assume
that p = Pn satisfies condition (1). Since <po 8;;-^1 is unital and self-adjoint,
it can be perturbed to a u.c.p. map 'ljJ by Corollary B.9. Extending 'ljJ to all
of pA**p, by Arveson's Extension Theorem, we are done. D