1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

96 3. Tensor Products

Now let's give an example where a tensor product sequence fails to be

exact.^22 Though elementary, the proof is somewhat delicate and requires a

few preliminary facts.

A discrete group r is called residually finite if there exist subgroups

r ~ r1 ~ r2 ~ ... such that each ri is a finite-index, normal subgroup of

r and nn r n = { e} (the neutral element of r). The key fact is that if r

is residually finite, then the minimal tensor product of the full group C* -

algebras C* (r) 181 C* (r) has a very special state coming from the left and

right regular representations.

If r is any discrete group, then we can consider the product map

>. x p: C*(r) 0 C*(r) __, JIB(£^2 (r))

induced by the (commuting) left and right regular representations. If we

take some elements x = L:::sEriYss, y = L:::tErf3tt E C*(r) (only finitely

many coefficients nonzero), then a straightforward computation shows that

(>. X p(x 181 y)De, De) = L 1Ysf3s·

sEI'

Note that if r happened to be a finite group, then C*(r) 0C*(r) = C*(r) 181

C* (r) and hence we would have a state on the minimal tensor product

satisfying the formula above.

Lemma 3. 7.9. If I' is residually finite, then there exists a stateμ on C* (r)181

C* (r) such that for finite sums

x =Lass, y = Lf3tt E C*(r)

sEI' tEI'

we have

μ(x 181y)=L1Ysf3s·

sEI'

Proof. Since each r /r n is a finite group, we have a sequence of states μn

on C* (r /r n) 181 C* (r /r n) satisfying the formula above (for r /r n). However

we also have quotient mappings 7rn: C*(r) -t C*(I'/I'n) and hence tensor

product *-homomorphisms

Since the intersection of the r n's is just the neutral element, a straightfor-
ward computation shows that any cluster point of the states μno (7rn 1817rn)
must satisfy the right formula. (Hence there is a unique cluster point of the
μn°(1fn1811fn)'s.) D

(^22) In Chapter 6, when we come to Kirchberg's factorization property, we'll revisit these ideas.