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3.6. Inclusions and The Trick 85

Corollary 3.5.6. Assume e: A--+ C and O": B--+ D are c.c.p. maps and

en: A--+ C are c.c.p. maps converging toe in the point-norm topology. Then

and

en @ (} --+ e @ (}

in the point-norm topology as well.

Exercise

Exercise 3.5.1. Let <p: A --+ JJll,(7-i) and 'lj;: B --+ JJll,(1-i) be c.p. maps with

commuting ranges. Show that there exists a c.p. map

<p Xmax 'lj;: A ©max B --+ JJll,(J-i)

such that <p Xmax 'lj;(a@ b) = <p(a)'lj;(b) for all a EA and b EB.

Even for representations, the previous exercise fails when one tries to

replace the maximal norm with the minimal norm. See Exercise 3.6.3 for a

natural counterexample coming from discrete groups.

3.6. Inclusions and The Trick

In this section we discuss one of the important subtleties of C* -tensor prod-

ucts. We also introduce one of .the great tensor product tricks, a technique

so important that it should not be regarded as a trick, but rather The Trick.

The issue at hand is whether or not inclusions of C* -algebras give inclu-

sions of tensor products. So long as one stays at the algebraic (i.e., pre-C* -

algebra) level, nothing funny happens (Proposition 3.1.11). This simple fact

implies that spatial tensor products are also well behaved in this regard.

Proposition 3.6.1. If A C B and C are C* -algebras, then there is a natural

inclusion

A@C c B@C.

Proof. Perhaps we should first point out what this proposition is really
asserting. Since we have a natural algebraic inclusion
A0C cB0C,

one can ask what sort of norm we would get on A 8 C if we took the spatial
norm on B 8 C and restricted it. This proposition asserts that we just get
the spatial norm on A 8 C.

Having understood the meaning of the result, the proof is now an im-

mediate consequence of Proposition 3.3.11. D