7.5. External approximation 255A crucial remark is that we can even arrange a certain amount of com-
mutativity. Namely, since the von Neumann algebras generated by {U1, Vi}
and {U 2 , 1/2} commute, we may choose the paths of unitaries to live inside
these respective von Neumann algebras; hence we may assume that the op-
erators { u1(t), v1(t)} commute with {u2(t), v2(t)} for all t E [O, 1]. It follows
that the identity representation C*(lFn x lFn) <-+ JB(H) is homotopic to the
trivial representation into Cl?-i and we are done, by Proposition 7.3.5. D
7.5. External approximation
In this section we exhibit some external approximation properties enjoyed
by QD C-algebras. All of the results obtained are, in one way or another,
inspired by single operator theory. While this has certainly led to a better
understanding of QD C -algebras, it has also given back to single operator
theory, as we will see in Chapter 16.
Our first result is a simple generalization of an argument which Halmos
used in the single operator case.
Theorem 7.5.1 (Approximation by RFD algebras). Let 1i be separable
and A c JB(H) be a separable C -algebra. If A is a quasidiagonal set of
operators, then for each finite set ~ C A and c > 0 there exists a residually
finite-dimensional C-algebra BC JB(H) and a u.c.p. map : A-+ B such
that
(a) - a E JK(H)
for all a E A and
ll(a) - all < c
for all a E ~.
Proof. Let a 1 , a2, · · · be dense in A and, by Proposition 7.2.3, find increas-
ing finite-rank projections, tending strongly to the identity, such that
1
II [Pk, aj] II < 2 kfor all k E N and 1 ::::; j ::::; k. If E; > 0 and a finite set ~ C A from the unit
ball are also given, we may assume that II [Pk, a] II < {k for all k and a E ~.
Consider the orthogonal finite-rank projections
Qi= P1,Q2 = P2 - P1,Q3 = P3 -P2, ....Since I: Qi= 1, we get a well-defined u.c.p. map by defining
00
(a) = L QiaQi,
i=l