74 3. Tensor ProductsThe remainder of this section is devoted to resolving the following tech-
nical issues.
(1) Are // · I/max and I/ · I/min norms (as opposed to seminorms)?^11
(2) Is I/ · I/min independent of the choice of faithful representations?
(3) Can one usually reduce the nonunital case to the unital case?
All three questions have affirmative answers, though none are completely
obvious.
Let us first tackle the norm vs. seminorm question. The following uni-
versal property of I/ · //max implies that it suffices to show I/ · I/min is a norm.
Proposition 3.3.7 (Universality). Ifw: AGE--+ C is a *-homomorphism,
then there exists a unique *-homomorphism A ®max E--+ C which extends
Jr. In particular, any pair of *-homomorphisms with commuting ranges
1f'A: A--+ C and 1f'B: E--+ C induces a unique *-homomorphism
1f'A X 1f'B: A®maxE-+ C.Proof. Faithfully representing C on some Hilbert space, this fact follows
from the definition of I/ · I/max· D
Corollary 3.3.8. The norm I/ · I/max is the largest possible C* -norm on
AGE.Proof. If I/· Ila is any other C-norm on AGE, then, by universality, there is
a (surjective) -homomorphism A®maxE--+ A@aE. Hence, I/xi/a::::; I/xi/max
for every x E A G E. D
In particular, I/· I/max dominates I/· I/min and thus, if I/xi/min= 0 =?-x = 0,
then it will follow that both I/ · I/max and I/ · I/min are honest norms.
Lemma 3.3.9. The product -homomorphism lBl(H) G lBl(K) --+ lBl(H ® K),
induced by the commuting -representations lBl(H) ~ lBl(H)@Clx: c lBl(H@K)
and lBl(K) ~ ClH ® lBl(K) C lBl(H ® K), is injective.
Proof. We must show that if a finite sum of tensor product operators
2-.:::i Si ® Ti E lBl(H ® K) is zero, then the corresponding sum of elemen-
tary tensors 2-.:::i Si® 'n E lBl(H) G lBl(K) is also zero. We may assume that
the operators {Si} c lBl(H) are linearly independent.
If 0 = 2-.:::i Si® 'n E lBl(H ® K), then for all vectors v, w E 1i and~' 17 EK
we have
((L Si® 'n)v ® ~' w ® 17) = 0.
i
(^11) Since both of these (semi)norms are defined via -representations and honest O-norms,
an affirmative answer to this question will imply that both 11 · llmax and II · llmin are 0*-norms.