1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

11.4. Excision and Papa's technique 329

where {QY)} are orthogonal projections and 1 = aii) > a~i) > · · · > aiili) >

0. The projections Qii) also excise rp; indeed, we have the following inequal-
   ity:

\\Qii)aQii) - rp(a)Qii)I\ = l\Qii)(eiaei - rp(a)e;)Qii)I\

:S \\eiaei - r.p(a)e;[[.

D

Note that a state rp can be excised by projections {Pi} if and only if
there exists a net of *-monomorphisms Pi: CC '----t A (i.e., a 1--+ api) such that

l\Pi(l)api(l) - Pi(rp(a))\\--+ O,

for all a E A. This somewhat awkward reformulation suggests how one
sho.uld generalize to u.c.p. maps with finite-dimensional range. Before stat-
ing the result, however, we need a basic perturbation fact: approximate
partial isometries with common support and orthogonal ranges can be per-
turbed to honest partial isometries with common support and orthogonal
ranges.

Lemma 11.4.3. For every c > 0 and n EN there exists 8 = 8(s, n) > 0 such
that if A is a C* -algebra, p E A is a projection and { w1, ... , wn} C A are
elements such that \wjwi - 8i,jP\ < 8 (8i,j is the Kronecker delta function),
then there exist partial isometries { v1, ... , vn} C A such that vjvi = 8i,jP
and \wi - vi\< c, for 1 :S i,j :Sn.

Proof. The proof is by induction so consider the case n = 1. Assume
we have w E A such that \ww - p\ < 8 < 1. Then [pwwp - p\ :S
[\ww - p\ < 8 and hence pwwp is a positive invertible element in pAp.
Applying functional calculus to pwwp with f(x) = x-^1!^2 , we can find
0 ::::; y E pAp such that ypwwpy = p and [\y - p\ :S ~ - 1. Defining
v = wpy, we have

\\wpy-w\\ = \\wp(y-p) + (wp-w)\\

1

:S \\w\\(ylf=8 -1) + \\(p-l)w*w(p-l)\\^112.

Since \(p - l)ww(p - 1)\^112 = \(p - l)(ww--: p)(p - 1)\^112 < 8112 and
\wl\ ::::; Vl+15, the right hand side of the inequality above goes to zero as
8--+ O, and this evidently proves then= 1 case of the lemma.

Now assume the lemma holds for n - 1, let c > 0 be given and let
{ w1, ... , wn} c A be such that \wjwi - bi,jP\ < 8. If 8 is sufficiently small,
we can apply the induction hypothesis to find partial isometries v1, ... , Vn-1