1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

334 11. Simple C* -Algebras

in the relevant inequality). Hence there is a maximal family {Ci hEI E S,

with units {qi}·

We show by contradiction that q = 'L,i qi = lM. So, assume not and let

p = 1 - q E M. Since pMp is injective, we can find a matrix subalgebra

B C pM p with unit e =/= 0 such that

llEB(exe) - (pxp - (p-e)x(p - e)) II~ :S c:^2 liell~-

To save space, we let D = B EB (f1i Ci)· Writing everything in matrix form

with respect to the decomposition 1 = q + e + (p - e), it is routine to verify

that

llEv((e + q)x(e + q))-(x - (p-e)x(p - e)) II~

= llEB(exe) - (pxp - (p - e)x(p-e)) II~

+ llErL ci (qxq) - (x - (1 - q)x(l - q)) II~

:S c

2

llell~ + c^2 llqll~

= c^2 lle + qll~-

Of course, this contradicts maximality of the family {Ci}, so q = lM.

Since 'L,iEI r(qi) = 1, we can find a finite subset Io c I such that

r(qo) > 1-E for qo = 'L,iEio qi. Finally, observe that C = CCqc} EB E9iEio Ci is

a finite-dimensional von Neumann subalgebra in M which almost contains

i. This proves that Mis AFD. D

Remark 11.4.9. For a while there was speculation that QD C-algebras
might always be nuclear. (There is even a "proof' of the false statement "QD
implies exact" in the literature.) A less ambitious question asks whether
every simple unital QD C -algebra with real rank zero is nuclear; Papa's
approximation property in Theorem 11.4.6 was viewed as evidence in favor
of this due to its similarity with the approximation property characterizing
the AFD II1-factor (Lemma 11.4.7). It turns out, however, that quasidi-
agonality and nuclearity are completely unrelated, even in the presence of
simplicity, real rank zero and almost any other assumptions one would like
to make. The original counterexamples were constructed by Dadarlat ([50]);
variations and refinements can be found in [30].

Also, we must remark that when the finite-dimensional algebras from
Theorem 11.4.6 can be taken "large in trace", the algebra becomes amenable
to classification. This fact was exploited in [65] and led Huaxin Lin to define
tracially AF algebras ([116]) and classify them ([117]).