332 11. Simple C* -Algebras
II [q, u] 112 = llquqj_ - qj_uqll^2
= max{llqJ_u*qll^2 , llqJ_uqll^2 }
= max{llquqJ_u*qll, llqu*qJ_uqll}
= max{llq - ququ*qll, liq - qu*quqll}
:S max{llq - Pi(<I?(u)<I?(u*))ll, llq - Pi(<I?(u*)<I?(u))ll}
+ 211 quq - Pi (<I? ( u)) II
= max{llP - P7rcp(u)P7rcp(u*)Pll, llP - P7rcp(u*)P7rcp(u)Pll}
+ 211 quq - Pi (<I? ( u)) II
= II [P, 7rcp( u)Jll^2 + 2llquq - Pi( <I?( u)) II·When <p is a trace, cptq) <po Pi is the tracial state on "IE(PHcp) and the proof
of the second inequality is similar:
II [q, uJll3,cp = llqJ_u*qll3,cp + llqJ_uqll3,cp
= <p(q - Pi(<I?(u)<I?(u*)) + q - Pi(<I?(u*)<I?(u)))
+ <fJ(Pi(<I?(u)<I?(u*)) - ququ*q + Pi(<I?(u*)<I?(u)) - qu*quq)
= <p(q) (tr (P - P7rcp(u)P7rcp(u*)P + P - P7rcp(u*)P7rcp(u)P))
+ <fJ(Pi(<I?(u)<I?(u*)) - ququ*q + Pi(<I?(u*)<I?(u)) - qu*quq):S llqll3,cp ( ll[7rcpli~ll{Jll^3 + 4llquq - Pi(<I?(u))ll),
where the last inequality uses the general Holder-type inequality l<p(ab)I :::;
<tJ(lal)llbll.
Theorem 11.4.6. Let A be simple unital quasidiagonal and let it have real
rank zero. For every finite set ~ C A and c: > 0 there exists a finite-
dimensional subalgebra B C A with unit Q such that II [a, QJll < c: and
d(QaQ,B) < c:, for all a E ~-Proof. Let <p be any state on A. Since A is simple and unital, we have that
the GNS representation 7r cp is faithful and its image contains no nonzero
compact operators. Since A is quasidiagonal, we can apply Theorem 7.2.5
to find a sequence of nonzero finite-rank projections Pn E "IE(Hcp) such that
II [Pn, 7rcp(a)Jll -+ 0 for all a EA. ·
By Proposition 11.4.2, <p can be excised by projections; hence we can ap-
ply Popa's local quantization technique to excise the maps a~ Pn7rcp(a)Pn.
Since the Pn's asymptotically commute, the commutator estimates evidently
imply the desired approximation property. D