228 6. Amenable TracesWith Theorem 6.2. 7 at our disposal, the following fact is immediate.
(See Exercise 6.1.4 for the identification with opposite algebras.)
Theorem 6.4.3. A discrete group r has the factorization property if and
only if the trace 7)., on C* (r) coming from the left regular representation is
amenable.Though it is a natural notion, relatively little is known about the fac-
torization property. It is clear that amenable groups enjoy this property
and Proposition 3.7.10 says that residually finite groups do as well.^12 Since
many classical groups are residually finite, this shows that we have lots of
examples with the factorization property. Unfortunately, there is only one
other nontrivial fact which is known about this concept, namely, that for
groups with Kazhdan's property (T), residual finiteness and the factoriza-
tion property are equivalent. This result, due to Kirchberg, will be our focus
for the remainder of this section.
Definition 6,.4.4. r has Kazhdan's property (T) if every unitary representa-
tion which has almost invariant vectors actually has invariant vectors. More
precisely, if 7r: r ----> IIB(H) is a unitary representation and Vi E 1{ is a net
of unit vectors such that I I 7r g (Vi) - Vi 11 ----> 0, for all g E r, then there exists
0 #-v E 1{ such that 7rg(v) = v for all g EI'.
Proposition 6.4.5. For a discrete group r the following statements are
equivalent:
(1) r has Kazhdan's property (T);
(2) (critical sets) there exists a finite set F c r and K, > 0 such that if
7r: r ----> IIB(H) is a representation and w E 1i is a unit vector such
that 117rg(w) - wll < K, for all g E F, then there exists a nonzero
vector vo E 1i such that 7r 8 ( vo) = vo for all s E r;
(3) (quantitative version) there exists a finite set F c r and K, > 0 such
that for every E > 0 and representation 7r: r ----> IIB(H), if w E 1{
is a vector such that 117rg(w) - wll < E for all g E F, then there
exists a vector vo E 1i such that 7r 8 (vo) = vo, for alls E r, and
llw - voll :S ~·
Proof. (1) =?- (2): Proving the contrapositive, we assume that for each
finite set F c r and K, > 0 there exists a representation 7r: r ----> IIB(H)
such that 7r has no fixed vectors but there is a unit vector v E 1{ such that
117rg(v) - vii < K, for all g E F. Applying this assumption to larger and
larger finite sets and smaller and smaller K,'s, we can find a collection of
(^12) Hence if r is a residually finite, nonamenable group, then the canonical trace on C* (r) is
amenable, but it is not amenable when regarded as a trace on C~(I').