76 3. Tensor ProductsBut since Pn'iPJ{H)Pn is naturally isomorphic to Mn(<C), we have
II L(Pn7r(ai)Pn)@ o-(bi)ll =II L(Pn7r(ai)Pn)@ cr'(bi)ll
for each n, since there is a unique C*-norm on Mn(<C) 0 B (Proposition
3.3.2). D
Finally we present a result which allows many nonunital questions to be
handled (relatively) painlessly. For a nonunital C*-algebra A we will let A
denote the unitization.
Corollary 3.3.12. If A is nonunital, then any C-norm II· Ila on A 0 B
can be extended to a C -norm on A 0 B. Hence, when both A and B are
nonunital, any C-norm II· Ila on A 0 B can be extended to a C-norm on
.A0-8.
Proof. Let II· Ila be a C-norm on A0B and 7f: A@aB -t JIB(1i) be a faithful
representation of the completion of A 0 B with respect to II· Ila· Let 7fA and
7fB be the restrictions given by Theorem 3.2.6. Since 7f is faithful, it follows
that 7f A is faithful and hence may be extended to a faithful -homomorphism
1f-A: A -t JIB(1i). Since the ranges of 1f-A and 7fB still commute, we can
consider the product map 1f-A x 7fB: A 0 B -t JIB(1i). If we knew that
1f-A x 7fB were injective, then we would be done since the norm on JIB(1i)
would restrict to a C*-norm on 1f-A x 1fB(A0B) ~ A0B which would agree
with II · Ila on A 0 B. If you didn't already do Exercise 3.1.6, now is the
ti~. D
Exercises
Exercise 3.3.1. Show that both 11 · llmin and 11 · llmax are commutative tensor
product norms - i.e., there are canonical isomorphisms A Q9 B ~ B Q9 A and
A ®max B ~ B ®max A.
Exercise 3.3.2. Show that both II · llmin and II · llmax are associative -
i.e., there are canonical isomorphisms (A Q9 B) Q9 C ~ A Q9 (B Q9 C) and
(A ®max B) ®max C ~ A ®max (B ®max C). How would you define the
maximal or minimal tensor product of n algebras?
Exercise 3.3.3. Give an example of a *-representation 7f: A 0 B -t JIB(1i)
such that both 7fA and 7fB are injective but 7f is not. (Hint: Think finite
dimensional and abelian.)
Exercise 3.3.4. Prove that if 7f: A -t JIB(1i) and er: B -t JIB(!C) are arbitrary
(not necessarily faithful) representations, then there exists a unique extend-
ing *-homomorphism 7f Q9 er: A Q9 B -t JIB(1i Q9 IC) such that 7f Q9 er( a Q9 b) =
7r( a) Q9 cr(b). (Hint: Dilate 7f and er to faithful representations and then cut
back down; see Exercise 3.2.2.)