1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

256 7. Quasidiagonal C* -Algebras

for all a E A. Letting B = C*(<I>(A)), it is clear that B is RFD. The

remainder of the proof is contained in the following calculation:

00 00

<I>(a) - a= L QiaQi - (L Qi)a

i=l i=l

00

i=l

00

i=l

Indeed, for arbitrary a E A all of the series above converge in the strong

operator topology. But for each aj the last sum is a norm convergent series

of finite-rank operators and thus, by continuity, <I>(a) - a E OC(1i) for all

a EA. D

Note that A+ OC(1i) = B + OC(1i) (since (a) - a E OC(1i) for all a EA)
and hence the images of A and B down in the Calkin algebra agree. In
particular, if An OC(1i) = {O}, then A is a quotient of B.
Corollary 7.5.2. Let A be separable 1 QD and let 7r: A-+ JB5(1i) be a faithful
essential representation. Then 1 there exists an RFD algebra B C n(A) +
OC(1i) and a short exact sequence
0 -+ OC(1i) -+ B + OC(1i) -+ A -+ 0
such that 7r provides a (-homomorphic) splitting.
Remark 7.5.3 (Fredholm index obstruction). Assume TE JB5(1i) is a Fred-
holm, quasidiagonal operator. Then Ind(T) = 0. Hence a Fredholm oper-
ator with nontrivial index can't be quasidiagonal. To see this, we use the
approximation result above. For operators on a finite-dimensional space,
the dimension of the kernel and cokernel are always equal; hence the same
holds for any Fredholm, block diagonal operator 8 = 81 E9 82 E9 · · ·. But if
Tis quasidiagonal, then C(T) c JB5(1i) is a quasidiagonal set of operators,
so T = (T) + k, where k is compact and (T) is block diagonal. Since
Ind(·) is invariant under compact perturbations, the claim is proved.
Remark 7.5.4 (Nonquasidiagonal representations). The previous remark
implies that if A is QD and 7r: A -+ JB5(1i) is a faithful representation such
that Ind(n(a)) -/= 0, for some a E A, then 7r can't be a quasidiagonal
representation. With this observation it is easy to construct examples of
QD C -algebras with nonquasidiagonal faithful representations. For exam-
ple, let 8 E JB5(£^2 (N)) be the unilateral shift and let A = C(8 E9 8) C
JB5(£^2 (N) E9 £^2 (N)). The operator 8 E9 S is a compact (actually, rank one)
perturbation of the (unitary) bilateral shift on JB5(£^2 (Z)); hence A is QD (see