1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

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   7. Exact sequences 97
   
   
   
   

Proposition 3.7.10. If r is residually finite, then the product map

,\ x p: C(r) 0 C(r)-+ JB(£^2 (r))

induced by the left and right regular representations is continuous with re-
spect to the spatial tensor product norm. (Compare with Exercise 3. 6. 3.)

Proof. Let ?T: C(r) ® C(r) -+ JB('H) be the GNS representation with
respect to the state constructed in the previous lemma. Uniqueness of GNS
representations implies that the (algebraic) representations

and

,\ x p: C(r) 0 C(r)-+ JB(£^2 (r))

are unitarily equivalent since Oe E £^2 (r) is a cyclic vector for the algebra
,\ x p( C (r) 0 C (r)) whose corresponding vector functional agrees with μ.
This implies the C -algebra generated by ,\ x p( C (r) 0 C (r) )-is a quotient
of C (I') ® C* (r), so the proof is complete. D

Proposition 3.7.11. Let r be a residually finite discrete group. Then the
following are equivalent:

(1) r is amenable;

(2) C*(I') is exact;

(3) the sequence

o-+ J 0 C(r)-+ C(r) 0 C(r)-+ C{(r) 0 c(r)-+ o

is exact, where J is the kernel of the quotient map C* (r) -+ C{ (r).

Proof. (1) ==?- (2) follows from Theorem 2.6.8 since full and reduced C*-
algebras of amenable groups always agree.

(2) ==?-(3) is a consequence of Proposition 3.7.8.

(3) ==?-(1): If

C* (r) ® C* (r) ~ C* (r) ® c* (r)

J®C*(I') >- '

then Proposition 3.7.10 implies that we have a -homomorphism ?T: C{(I')®
C(r) -+ JB(£^2 (r)) such that ?T(x ® y) = xp(y) for all x E C{(I') and
y E C (r). Indeed, we know that the product map ,\ x p extends to a
-homomorphism which we will denote by

,\ Xmin p: C(r) ® C(r)-+ JB(£^2 (r)).