328 11. Simple C* -Algebras
Proof. Simple QD algebras are inner QD and hence the assumption of
nuclearity implies that A must be strong NF. The conclusion then follows
from Proposition 11.2.8. D
11.4. Excision and Popa's techniqueTheorem 1.4.10 states that pure states can be excised. In this section we
observe that a similar result holds for certain finite-dimensional u.c.p. maps,
when the domain algebra has a sufficient supply of projections. We then use
Popa's technique to derive some important corollaries.
Definition 11.4.1. Let A be a C*-algebra and <p E S(A) be a state on
A. We say that <p can be excised if there exists a net of positive, norm-one
elements ei EA such that f[eiaei - <p(a)erfl--+ 0 for all a EA.
If one can choose the e/s to be projections, then we say <p can be excised
by projections.Recall that a C* -algebra A has real rank zero if every self-adjoint element
in A can be approximated by a self-adjoint element with finite spectrum -
in particular, real rank zero implies a rich supply of projections. The results
of this section hold under weaker hypotheses regarding the abundance of
projections, but we'll stick to the real rank-zero context as it is sufficient for
most applications. (See [159] for more general results.)
Proposition 11.4.2. Let A be a simple unital infinite-dimensional C* -
algebra with real rank zero. Then every state on A can be excised by projec-
tions.Proof. First let's observe that every state can be excised. For this, it suffices
to show that the pure states are weak-* dense in the state space of A.
However, this fact follows easily from Glimm's Lemma (Lemma 1.4.11) and
the simplicity of A. Indeed, if A c JIB(H) is an irreducible representation,
then A n IK(Jl) = 0, since A is unital and simple, and thus every state on
A can be approximated by vector states (irreducibility implies vector states
restrict to pure states on A); hence, every state on A can be excised.
We now use real rank zero to excise by projections. Fix a state <p on A
and let ei be a net of positive, norm-one elements in A such that [[eiaei -
<p(a)erfl--+ 0 for all a EA. Perturbing a little, we may assume that each ei
has finite spectrum and hence we can write
k(i)
ei = .l:a)i)Qy),
j=l