86 3. Tensor ProductsApplying this fact again on the right hand side implies that a pair of
inclusions Ac Band Cc D induces a natural inclusion A Q9 CC B Q9 D.
For maximal tensor products the question then becomes: If A C B
and C are given, do we have a natural inclusion A ®max C C B ®max C?
In general this turns out to be false and may seem a little puzzling at first.
However, when reformulated at the algebraic level, it becomes clear what can
go wrong. Indeed, what we are really asking is whether or not the maximal
norm on B 0 C restricts to the maximal norm on A 0 C C B 0 C. But
the maximal norm is defined via a supremum over representations and since
every representation of B 0 C gives a representation of the smaller algebra
A 0 C, it is clear that the supremum only over representations of B 0 C
will always be less than or equal to the supremum over all representations
of A0C.
Having seen what the problem could be, it's not too hard to formulate
a condition which ensures that inclusions behave nicely for maximal tensor
products.
Proposition 3.6.2. Let A c B be an inclusion of C* -algebras and assume
that for every nondegenerate *-homomorphism 7r: A ----"* lffi('h'.) there exists a
c.c.p. map <p: B----"* 7r(A)" such that <p(a) = 7r(a) for all a EA. Then for
every C* -algebra C there is a natural inclusion
A ®max C C B ®max C.Proof. By universality, we have a canonical -homomorphism A ®max C----"
B ®max C. Our goal is to show that if x E A ®max C is in the kernel of this
map, then x = 0.
Let 7r: A ®max C ----" lffi('h'.) be a faithful representation and 7rA: A ----"
lffi('h'.), 7rC: C ----" lffi('h'.) be the restrictions given by Theorem 3.2.6. Note
that 7rc(C) C 7rA(A)' and hence the commuting inclusions 7rA(A)" '----1-lffi('h'.),
7rc(C) '---7 lffi('h'.) induce, by universality, a product -homomorphism
7rA(A)" ®max 7rc(C) --+ lffi('h'.).Extend nA to a c.c.p. map <p: B----"* 7r(A)" such that <p(a) = 7r(a) for all
a E A. By Theorem 3.5.3 we have the following commutative diagram:
B ®max C 'P®max7rO 7rA(A)" ®max 7rc(C)I l
A ®max C ___ 7r ___ _,,__ lffi('h'.).The fact that 7r is faithful implies that the map on the .left is also injective.
D