6.4. Factorization and property (T) 227Exercise 6.3.3. Prove that r is amenable if and only if C!(r) has a finite-
dimensional representation.6.4. The factorization property and Kazhdan's property (T)
Before defining Kirchberg's factorization property, let's make some observa-
tions which help motivate the concept.
In Exercise 3.6.3 we observed that a discrete group r is amenable if and
only if the product map,
CHr) 0 c;(r)-+ JIB(£^2 (r)),induced by the left and right regular representations is min-continuous. One
might wonder what would happen if the reduced group C* -algebra were
replaced by the universal one.
Proposition 6.4.1. If r is a discrete group, the following are equivalent:
(1) r is amenable;
(2) the product *-homomorphism C!(r) 8 c;(r) -+ JIB(£^2 (r)) is min-
continuous;
(3) the product *-homomorphism C*(r) 8 c;(r) -+ JIB(£^2 (r)) is min-
continuous;
(4) the product *-homomorphism C!(r) 8 C*(r) -+ JIB(£^2 (r)) is min-
continuous.Proof. We only need to show (3) =} (1) and (4) =} (1), but they are similar
so let's do (3) =} (1). Applying The Trick to the inclusion C(r) 0 c;(r) c
C(r) 0IBl(£^2 (r)), we get a u.c.p. map from JIB(£^2 (r)) into the von Neumann
algebra generated by the right regular representation which restricts to the
identity on the image of the right regular representation. As we have seen
(in the proof of Theorem 2.6.8), this implies the existence of an amenable
trace on C!(r), so the proof is complete. D
Of course, if spatial tensor products are replaced by maximal ones, then
continuity is never an issue. Hence, if we want a nontrivial condition which
doesn't imply amenability, then we are forced to consider the spatial tensor
product with the universal group C* -algebra in both variables.
Definition 6.4.2. A discrete group r has Kirchberg's factorization property
if the product map,
C(r) 0 C(r) -+ JIB(£^2 (r))
induced by the left and right regular representations is min-continuous.^11
l11n other words, the product map C(r) ©maxC(r) -+ lffi(£^2 (r)) factors through the spatial
tensor product.