1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

10.2. Exact C* -algebras 305

Proposition 10.2.8. For any discrete group I', the C*-algebra

C*({Ag 0 Ag: g EI'}) c Ct(I') 0max C~(r)

is isomorphic to C* (I'). In particular, C~ (IF 2) 0max C~ (IF 2 ) is not exact since

it contains a nonexact subalgebra.

Proof. Let O": C* (r) --+ C~ (I') 0max C~ (r) be the *-homomorphism induced

by the map g 1--+ Ag 0 Ag. Fix a faithful representation ?r: C*(r) --+ IBl(H).

Our goal is to show

llK(x)ll:::; 1IO"(x)JI

for all x E C*(r), as this.implies O" is injective.

The trick is to construct a pair of commuting representations of r such

that 7f is a subrepresentation of the product. Fell's absorbtion principle

makes this straightforward. Indeed, we have a representation

and a commuting representation given by

Ag 1--+ Pg 0 lrt·

Hence we have a product representation

with the property that

(-) o O"(g) = AgPg 0 Kg.

The subspace Oe 01i is evidently invariant for each unitary e o O"(g ), and the

restriction of e o O" to Oe 0 1i is 1f. This implies the desired inequality. 0

This result suggests the following conjecture: If A and B are exact C* -
algebras, then A 0max Bis exact if and only if either A or Bis nuclear.

Crossed products. There are many characterizations of exact groups -see

Section 5.1. One of them is relevant to crossed products (Theorem 5.1.10).

Theorem 10.2.9. Let r be an exact group and A be a I'-C*-algebra. Then
A ><lr r is exact if and only if A is exact.

Proof. The "only if' direction is trivial. For the converse, let 0 -+ J --+
B -+ (B / J) -+ 0 be short exact and note that 0 -+ J 0 A -+ B 0 A -+
( B / J) 0 A --+ 0 can be regarded as a short exact sequence of r-algebras by
tensoring with the trivial action of r on B (compare with the first part of