1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.6. Inclusions and The Trick 91

for all a E A, c E 0. Evidently \If is a *-homomorphism and it is injective
since this is the case when restricted to A 0 C (Exercise 3.4.1). D

The reader who is really paying attention may have noticed a glaring
contradiction at this point. On the one hand, we already observed that
inclusions behave nicely when the subalgebra is nuclear (Corollary 3.6.3).
However, we also claimed that the previous proposition shows how badly
inclusions can behave for exact C -algebras. But nuclear C -algebras are
also exact, so how can this be?

Proposition 3.6.12.^16 If A is nuclear, then for every C -algebra C there is
a unique C-norm on A00. In other words, the canonical quotient mapping
A@max C ~A@ C
is injective.

Proof. Apply Lemma 3.6.10 to (} = idA: A ~ A. D

Exercises

Exercise 3.6.1. Why is Corollary 3.6.3 an immediate consequence of Propo-
sition 3.6.12?

Exercise 3.6.2. Assume A has the WEP and let 1f: A ~ IIB('H) be a non-
degenerate faithful^17 representation. Prove the existence of a u.c.p. map
cI>: IIB(H) ~ 7r(A)" such that <P(7r(a)) = 7r(a) for all a EA. (There are two
simple proofs: one using the definition of the WEP and Arveson's Extension
Theorem and another based on The Trick.)

Exercise 3.6.3. Let r be a discrete group and let

. x p: o~(r) 0 o;(r) ~ IIB(e^2 (r))

be the product of the left and right regular representations. Prove that r
is amenable if and only if >. x p is continuous with respect to the minimal
tensor product norm.

Exercise 3.6.4. Show that if A is nuclear and 1f: A ~ IIB(H) is any nonde-
generate representation, then 7r(A)' is injective. (Hint: If it takes you more
than two lines, then you have missed the point! The Trick is your friend.)

Exercise 3.6.5. Let X be a locally compact Hausdorff space and Co(X)
be the continuous functions vanishing at oo. For a C* -algebra A we let

Co(X,A) = {f: X ~A: f is continuous and f(oo) = O}.

16This is also the easy direction of a hard result; see Theorem 3.8.7.
17 This exercise can fail for noninjective representations. Giving counterexamples isn't trivial,
though.