1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

98 3. Tensor Products

However, it is evident that J 0 C*(r) belongs to the kernel of,\ Xmin p and

hence this *-homomorphism factors through

C* (r) 0 C* (r) ~ C* (r) 0 C* (r)

JQ9C*(r) >. '

so we get our representation 7r.

To finish off the proof, use The Trick to get a u.c.p. map <P: JIB(£^2 (r)) ____,

L(r) such that <I>(x) = x for all x E C~(r). This is very similar to the proof

of Proposition 3.6.9 (with maximal norms replaced by minimal ones and the

right regular representation regarded as originating from C* (r)), so we leave

~~~~~. D

As promised in the last chapter, we finally prove the existence of nonex-

act C* -algebras. This follows immediately from the previous result together

with Proposition 3. 7.8.

Corollary 3.7.12. If r is a nonamenable residually finite group {e.g., IB'n

or SL(n, Z)), then C*(r) is not exact since the sequence

o -> J 0 C*(r) ____, C*(r) 0 C*(r) ____, CHr) 0 C*(r) ____, o

is not exact.

It follows from this corollary that JIB(C^2 ) is not exact either (since exact-

ness passes to subalgebras).

Exercises

Exercise 3."f.1. Use Lemma 3.7.7 to show that if the sequence

O ____, J 0 C ____, A 0 C ____, A/ J 0 C ____, O

is exact and B c C is a C* -subalgebra, then

0 ____, J 0 B ____, A 0 B ____, A/ J 0 B ____, 0

is also exact.

Exercise 3!l.2 (Uniqueness of the extension in Corollary 3.3.12). Let a

and f3 be C*-norms on A8B, where A is the unitization of A. Prove that if

a= f3 on A 8 B, then a= f3 on A 8 B. In particular if A 8 B has a unique

C*-norm, then so does A0B. Hint: Consider ry = max{a,/3} and use the 5

Lemma^23 on the following diagram:

0 -----+ A0ryB --1 A 0ry B -----+ CCl 0 B -----+ 0

l l l

0 -----+ A !Zia B -----+ A !Zia B -----+ CCl 0 B -----+ 0.

(^23) See Mathworld (mathworld.wolfram.com/FiveLemma.html) if you're not familiar with the
5 Lemma.