88 3. Tensor Products
and u.c.p. maps are bimodule maps over their multiplicative domains, a
simple calculation completes the proof.
The nonunital case is a bit more irritating but can be deduced from the
unital case as follows. For a C -algebra D, let fJ be the unitization if D
is nonunital and fJ = D if D is already unital. For an arbitrary inclusion
AC Band auxiliary algebra C we may extend any C-norm II· Ila on EGG
to unitizations (Corollary 3.3.12) and get an inclusion B 0a C C B 0a 6.
Let Ai =A+ Cl13 (which may or may not be the same as A) and note that
Ai 8 C c B 0a 6. Hence II · 11.B extends to a norm which yields an inclusion
A 0(3 C c Ai 0,B 6. The key observation is that A 0(3 C is an ideal in
Ai 0,B 6 and hence any representation of A0,B C extends to a representation
of Ai 0(3 C. Given this fact, it is easy to deduce the general case from the
unital one proved above. 0
At first glance, the utility of The Trick is far from obvious, but please
be patient as the mileage one can get out of this simple observation is re-
markable. Let us briefly explain what the point is and then we will give an
application.
Given an inclusion AC Band a representation 7r: A-+ JIB(H), Arveson's
Extension Theorem always allows one to extend Jr to a c.c.p. map <p: B -+
JIB(H). When The Trick is applicable, it gives one the ability to better control
the range of <p and this is how it gets used. As our first example we provide
the converse of Proposition 3.6.2, promised earlier. An inclusion satisfying
one of the following equivalent conditions is called relatively weakly injective.
Proposition 3.6.6. Let A C B be an inclusion. Then the following are
equivalent:
(1) there exists a c.c.p. map <p: B -+A* such that <p(a) =a for all
a EA;
(2) for every -homomorphism 7r: A -+ JIB(H) there exists a c. c.p. map
<p: B-+ 7r(A)" such that <p(a) = 7r(a) for all a EA;
(3) for every C* -algebra C there is a natural inclusion
A 0rnax C C B 0rnax C.Proof. Since every representation of A extends to a normal representation
of A**, the equivalence of the first two statements is easy.
Assume condition (3) and let 7r: A -+ JIB(H) be a representation. Let
C = 7r(A)' and, by universality, we can apply The Trick to the product
map induced by the commuting representations Jr: A-+ JIB(H) and 7r(A)' <----+
JIB(H). That's it. O