- Exact sequences 93
Did you catch the subtlety? Since J ©max B is much bigger than J 8 B,
is it not possible that (J ©max B) n (A 8 B) contains more than just J 8 B?
This would imply that the map
---+----A8B A@maxB
J0B J©maxBis not injective and we would have a fatal gap in the proof. Luckily the proof
is correct as the fact that (A/ J)8B sits faithfully inside (A/ J)©maxB (which
is a quotient of 1::::~) ensures that the map
---+----A0B A@maxB
J0B J©maxBis injective. D
The reader who understood the previous proof should have no trouble
demonstrating the following fact. In the literature this result is used fre-
quently and without reference -we will do the same.
Proposition 3.7.2. Given J<JA and B, there is a C*-norm II· Ila on (A/ J)8
B such that
A@B rv
J©B = (A/J)©aB.
Moreover, the sequence
O-+ J © B-+ A@ B-+ (A/ J) © B-+ O
is exact if and only if II · Ila is the spatial norm.
It is only slightly harder to give an example of a nonexact tensor product
sequence than it is to give an example where inclusions of maximal tensor
products fail. We will do this at the end of the section, but let's first consider
a number of cases where things go well. The following is a special case of
the previous result.
Corollary 3. 7.3. If there exists a unique C* -norm on (A/ J) 8 B, then the
sequence
0-+ J © B-+ A© B-+ (A/ J) © B-+ 0
is exact.The next fact follows from Corollary 3.7.3 and Proposition 3.6.12.Corollary 3. 7.4. If either A/ J or B is nuclear, then
0-+ J © B-+ A @B-+ (A/ J) © B-+ 0
is exact.