9.2. Tensor product properties 289B.14). Passing to a convex combination if necessary, we may assume that
the net {cpi(l)} converges in norm to the unit. Hence, we can perturb each
Soi to a u.c.p. map, thanks to Corollary B.11. D
Lemma 9.2.6. If I <l C and x EI** n C, then x EI.Proof. The canonical map C** -+ ( C /I)** restricted to C is just the quo-
tient map C -+ C /I. Hence, the set of elements in C belonging to the kernel
is equal to I. On the other hand, the kernel of C** -+ ( C /I)** is I**, which
evidently implies the lemma. D
Proposition 9.2. 7. A C* -algebra is exact if and only if it has property C'.Proof. We first assume A has property C' and let B be an arbitrary C* -
algebra with ideal J <l B. Since 0 -+ J -+ B -+ (B / J)** -+ 0 splits,
Proposition 3.7.6 implies that
0 -------+ J** ©A -------+ B** ©A -------+ (BI J)** ©A -------+ 0
is exact. Since we have assumed property C', we get a natural commutative
diagram
0 -------+ J©A -------+ B©A -------+ (B/J) ©A -------+ 01 1 1
0 -------+ J** ©A -------+ B**©A -------+ (B/J)** ©A -------+^01 1 1
0 -------+ (J ©A)** -------+ (B ©A)** -------+ ((B/J)©A)** -------+ o.It is important to remember that at this point we do not know if the bottom
row is exact (since we don't know if the top row is exact). But, if we take x E
B@A with the property that xis in the kernel of the map B@A -+ (B / J)©A,
then exactness of the middle row implies that x E ( J ©A) n B ©A = J ©A
(by Lemma 9.2.6). This proves that A is exact.
For the converse, assume A is exact and B is arbitrary. For a directed
set I, we let
B1 = {(x(i))iEI E IT B: strong*-~~x(i) exists in B*}.^1
iEJ
It is easily checked that B1 is a C -subalgebra of IJiEI B (multiplication
on bounded sets is jointly strong -continuous). Evidently we have a -
homomorphism CT: BI -+ B, given by
(x(i))iEI f---7 strong*-~~1]1-x(i) EB**,
lThis means the usual strong*-topology in the universal representation B** C JIB('h'.u): Ti-+
T strong* if and only if Ti -+ T and Tt -+ T* in the strong operator topology.