1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

440 17. K-Homology

Proof. Let (j;: A---+1IB(1i) be a u.c.p. lifting and o-: A-+ 1IB(1i) be a faithful

unital essential *-representation. Since <j;(A) is a quasidiagonal set, there is

a sequence of finite-rank projections Pn which converges to 1 strongly and

commutes with <j;(A) asymptotically. Consider the u.c.p. maps <Pn: A ---+

lIB(PtH) obtained by compressing <p to PtH. Clearly all the rfn's still

represent [<p] (after passing to the Calkin algebra) and it is easy to see that

the sequence ( rfn) is asymptotically multiplicative. Hence by Theorem 1. 7.6,

there are unitary operators Un: PtH---+ K such that

lim [[ Adun orpn(a) - o-(a)[[ = 0

n-+oo

for all a E A. This means that d([<p], [o-]) = 0. D

If you actually checked that our notion of quasidiagonality for elements

of Extw is well-defined, you may also want to prove the converse of the

previous lemma for QD C*-algebras ([171, Theorem 2.9]).

Let (u1(n), ... ,uk(n)) E MN(n)(C)k be a coding family of unitary k-

tuples (Definition 13.5.3), which means that

k

sup{[['°"' L.J ui(m) Q9 ui(n)flrur N(m) (iC)""M "" N(n) (IC) : m #-n} = k - 5

i=l

for some 5 > 0. Let

00 00

n=l n=l

and define A to be the C*-algebra generated by {v1, ... , vk} in Q(ffi£7-.r(n)).

Proposition 17.4.3. Let A be as above. Then 1 the set of quasidiagonal

elements in Extw(A) is nonseparable.

Proof. Passing to a subsequence, we may assume that the sequence

[[f(u1(n), ... , uk(n))[[

is convergent for every f E C[IB'k]· For an infinite subset a c N, we identify
ITnEa MN(n)(q with the corresponding block-diagonal subalgebra of:IIB(Ha),

where 1-ia = EBnEa £7-.r(n). Let

ui = (ui(n))nEa E 1IB(1-ia) and vf = ui + lK(1ia) E Q(1ia)·

Observe that [[f(vf, ... , vk) [[ = limn IB'k, ... , uk(n))[[ for every f E
C[IB'k], and hence 'Pa(vi) = vf defines a *-monomorphism 'Pa from A into
Q(1ia)·