288 9. Local ReflexivityIt turns out that property C' also passes to quotients (remarkably, prop-
erties C and C' are equivalent) but the proof above cannot be adapted to
this case.
Coming back to finite-dimensional approximation properties, the next
two results are very important. We will need to use operator space duality
so the reader may wish to review Theorem B.13 before reading the proof of
the next proposition.
Proposition 9.2.5. A C* -algebra is locally reflexive if and only it has prop-
erty C".Proof. Suppose A is locally reflexive and let B be an arbitrary C* -algebra.
It suffices to show JJzJJ(A®B)** ~ JlzllA**®B for a given z = I:~=l ak 0 bk E
A** 0 B. Let E be the operator system in A** spanned by ai, ... , an. Since
A is locally reflexive, there exists a net of c.c.p. maps 'Pi: E --+ A which
converge to idE in the point-ultraweak topology. Because A** 0 B** --+
(A@B)** is bi-normal and multiplication by a fixed operator is ultraweakly
continuous, it follows that for every x E E and b E B we have
'Pi(x)@ b--+ xb E (A@ B)**
in the ultraweak topology on (A@ B)**. This implies that
('Pi 0 idB)(z)--+ z E (A 0 B)**
in the ultraweak topology on (A@B)**. Hence we get the following inequal-
ities:We now prove the "if" direction, so assume A has property C". We
may assume that A is unital. Let E C A be a finite-dimensional opera-
tor system. Invoking operator space duality (Theorem B.13), the inclusion
E '---+A corresponds to an element z EE*@ A* with JJzll = 1. We may
assume that E C IIB(H) completely isometrically. Consider the following
commutative diagram of canonical inclusions:
E' r" ---~ (E' f A)"
IIB(H)@ A (IIB(H) @A).
Since the vertical inclusions and the bottom inclusion are all isometric, so
is the top inclusion. Therefore, we have Jlzll (E*®A)* = 1. It follows that
there exists a net {zi} in E@ A such that supi Jlzill ~ 1 and the net {zi}
converges to z in the weak-topology on (E @A)**. Applying operator
space duality again, each Zi corresponds to a c.c. map 'Pi: E --+ A such that
the net {'Pi} converges to idE in the point-ultraweak topology (see Remark