8 1. Fundamental Facts
and 7r(a^112 )Q = 7r(bi)Q, for all i EI. Since a and Q commute, one has
7r(br)Q = 7r(bi)(7r(a^1 l^2 )Q) = (7r(bi)Q)7r(a^1 l^2 ) = n-(a)Q.
Hence Ci = br satisfy the required conditions. D
Lemma 1.4.9. Let A be a C -algebra, <p be a pure state and L = {a E A :
<p( aa) = O} be the associated left ideal. Then we have ker <p = L + L *.
Proof. Let x E ker r.p be nonzero. Let ( 7r, H, e) be the GNS triplet and K
be the 2-dimensional subspace spanned bye and 7r(x)e. Since e l 7r(x)e,
Kadison's Transitivity Theorem provides us with a positive element b EA
such that 7r(b)e = e and 7r(b)(7r(x)e) = 0. It follows that bx E L and
(1-b)xEL*. D
Here is half of the Akemann-Anderson-Pedersen Excision Theorem (see
[1] for the other half).
Theorem 1.4.10 (Excision). Let A be a C*-algebra and <p be a pure state.
There exists a net (ei) CA such that 0:::; ei:::; 1, r.p(ei) = 1 and limi lleiaei-
1.p(a)erll = O for every a EA.
Proof. We first assume that A is unital. Let L be the left ideal associated
to r.p and (ci) be a right approximate unit for L (i.e., (ci) is an approximate
unit for the hereditary subalgebra L n L and for every a E L we have
Ila - acill --+ 0). Let ei = 1 - Ci. Since a - 1.p(a) E ker<p = L + L, we have
lim llei(a - r.p(a))eill = 0 and r.p(ei) = 1.
Now suppose that A is nonunital and take ei as above for A. Let (bj) be a
quasicentral approximate unit for A such that 1.p(bj) = 1. (Existence follows
from Kadison's Transitivity Theorem.) Then, bjeibj does the job. D
Here is a nonstandard proof of a fundamental fact.Lemma 1.4.11 (Glimm's Lemma). LetA C lffi(H) be a separable C*-algebra
containing no nonzero compact operators on H. If <p is a state on A, then
there. exist orthonormal vectors (en) such that wen (a) --+ t.p( a) for all a E A,
where wen (T) = \Ten, en).
Proof.^3 Let i CA be a finite subset of norm-one elements, c: > O, Koc 1t
be finite dimensional and PK, 0 be the orthogonal projection onto K 0. It
suffices to show the existence of a unit vector e E K6- such that lwe(a) -
r.p(a)I < 6c: for all a E i. By the Krein-Milman Theorem, there exists a
convex combination 'I/; = L:k=l Ak'l/Jk of pure states 'l/Jk such that r.p R:j~, 10 'I/;.
By excision, for each k, there exists a norm-one positive element ek E A
(^3) we thank Akitaka Kishimoto for showing us this short proof.