1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

330 11. Simple C* -Algebras

such that llwi-vill :Sf/ (where 8' will be determined later) and vjvi = 8i,jP,

for 1 .::; i, j .::; n - 1. Letting Q = L::?-l Vivi, one sees that

n-l n-l

11(1-Q)wn -wnll :SL llviwnl/ :S L(llvi -willllwnll + 8)

1 1

.::; (n - 1)(8'V'1+J + 8).

The remainder of the proof is bookkeeping, nothing more; the point is

that (1 - Q)wn is almost a partial isometry with support p (since it's close

to wn), so we can perturb it to an honest partial isometry with support p

and range orthogonal to Q, by the (proof of the) n = 1 case. We leave the

details to the reader. D

We are now ready to excise certain u.c.p. maps. The excisable maps arise

as compressions in the GNS constructions of nice states. More precisely, let

cp E S(A) be a state which can be excised by projections and 7rcp: A-+ lIB(1-icp)

be the corresponding GNS representation. Let {Yi}~ 1 C A be such that

cp(yjyi) = 8i,j (i.e., in 1-icp, {Yi}~ 1 is an orthonormal set of vectors), let

PE JIB(1-icp) be the orthogonal projection onto the span of {Yi}~ 1 and define

a u.c.p. map <l?: A-+ PlIB(1-icp)P by <l?(a) = P7rcp(a)P.

Theorem 11.4.4 (Popa's local quantization). With notation as above, the

u.c.p. map <l? can be excised, meaning there exists a net of *-monomorphisms

Pi: PlIB(1-icp)P '---+A such that

llPi(P)api(P) - Pi(<l?(a))ll-+ 0,

for all a EA.

Proof. Let ~ C A be a finite set of norm-one elements which contains the
unit of A, and let E > 0 be arbitrary. We must prove the existence of a
*-monomorphism p: PlIB(1-icp)P '---+A such that llp(P)ap(P) - p(<l?(a))ll < E,
for all a E ~.

Since cp can be excised by projections, we can, for any 8 > 0, find a
projection p E A such that llP(YjXYi)P - cp(yjxyi)Pll < 8 for all x E ~ and
1 :S i,j :S m. In particular, note that ll(YjP)*(YiP) - 8i,jPll < 8 for all
1.::; i,j _::; m.

In other words, {YiP }~ 1 is almost a set of partial isometries with orthog-
onal ranges and common support p. Thus, if 8 is sufficiently small, we can
perturb them to honest partial isometries {vi} such that vjvi = 8i,jP and
llvi - YiPll < E/6m^2 (8 depends on m and E; however we may also assume
that 8 < E /2m^2 ). Hence if we define fi,j = vivj, then {fi,j} is a set of matrix
units for an m x m-matrix algebra. Moreover, if we let q = I:: fi,i be the