1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

424 16. Herrera's Approximation Problem

To ease notation, let B = 7r(A). We have a short exact sequence

0 ---+ lK ( 1i) ---+ A + lK( H) ---+ B ---+ O

and by the previous corollary we can find a u.c.p. map <I>: B ---+ A+ lK(H)

such that 7r o <I> = idB. Since A C JE(H) is a quasidiagonal set, we can find

increasing finite-rank projections Pn ~ Pn+l which tend to the identity and

asymptotically commute with A+ lK(H). To show that Bis QD, it suffices

to find asymptotically multiplicative, asymptotically isometric contractive

c.p. maps from B into A+ lK(H).

Let Prt" = 1 - Pn and consider the (nonunital) c.p. (isometric) maps

<I>n(b) = P;t-<I>(b)P;t-.

Note that <I>n(b) E A+ lK(1i) for all n and b E B, since Pn E A+ lK(H).

Thus the proof will be complete once we understand why

for all b, c E B. But this follows from the fact that { Pn} is an approximate

unit for lK(H) and quasicentral in A+ lK(H). D

Proposition 16.2.4. With the same assumptions as the previous proposi-

tion, for each finite set J C A and E > 0 there exists a finite-dimensional

C* -subalgebra D C Q ( H) such that 7r ( J) C^6 D - i.e., for each a E J there

exists d ED such that 117r(a) - dll < c:.

Proof. Let B = 7r(A) and <I>n(b) = Prt"<I>(b)Prt" as in ·the proof of the last

proposition. Since each <I>n is a splitting, we may regard <I>n as a faithful

*-homomorphism modulo the compacts (as in Theorem 1.7.6). To make <I>n

unital, we regard it as taking values in JE(Prt"H) (rather than JE(H)). Keep

in mind that Prt"H has finite codimension in 1i and hence the Calkin algebra

can't see the difference between the two.

Let <T: B ---+ JE(JC) be a faithful essential representation. Since B is exact

and QD it follows from Dadarlat's approximation theorem (Theorem 7.5.7)

that we can find finite-dimensional subalgebras in IB(JC) which approximate

prescribed finite subsets of <T(B) arbitrarily well. Since the maps <I>n: B ---+

IB(Prt"H) are asymptotically multiplicative faithful *-homomorphisms mod-

ulo the compacts, we can, by Theorem 1. 7.6, find unitary operators Un: JC ---+

Prt"H such that

for all b E B. So, if a finite subset of B and c: > 0 are given, we can
find finite-dimensional approximations inside Q(H) by. first approximating
in the representation <T: B ---+ IB(JC), then conjugating over to ~n: B ---+
IB(Prt"H) (for a sufficiently large n) and, finally, passing to Q(Prt"H). Since