12.1. Kazhdan's property (T) 343that if a convex combination e = I:~=l akek of unit vectors (ek) in 'H is
close to a unit vector 'T/, then each fa is close to 'T/· More precisely, we have
the estimate
n
2: akllTJ - ek11^2 = 2 - 2ar\TJ, e) = 2ar\TJ, 'T/ - e) :s; 2117/ - e11.
k=l
For convenience, we say a unitary representation (7r, 'H) of r is universal
if the induced representation of the full group C* -algebra C* (r) is faithful
(i.e., 1T weakly contains any other unitary representation). For instance, we
can take ( 1T, 'H) to be the direct sum of all cyclic unitary representations of
r.
Lemma 12.1.8. Let r be a group which is generated by a finite symmetric
set S and v: S ---+ IR be a strictly positive symmetric function (i.e.) v( s) =
v(s-^1 ) > 0 for every s ES). For a universal representation (1T, 'H) of I', set
lvl = l:sES v(s) and
1
h = ~ 2:v(s)?T(s) E IIB('H).
sES
Then) r has property (T) if and only if 1 is isolated in the spectrum CJ(h)
of the self-adjoint operator h. Moreover) if CJ(h) C [-1, 1 - c] U {1}; then
( S, .J22) is a K azhdan pair.Proof. First, observe the following consequences of uniform convexity: e E
'H is r-invariant if and only if it is an eigenvector of h with eigenvalue
1, and a sequence (en) of unit vectors is almost r-invariant if and only if
lim\hen, en) = 1.
Now, supposer has property (T). Since a universal representation weakly
contains the trivial representation, it must contain the trivial representation.
Thus 1 E CJ(h) and we must show it's isolated. Letting JC C 'H be the or-
thogonal complement of the I'-fixed vectors (which is a reducing subspace
for h), it suffices to show 1 is not in the spectrum of hlK:. But, if it were in
the spectrum, we could find unit vectors en E JC such that lim(hen, en) = 1;
this implies JC has almost invariant vectors, which is impossible. Hence, 1 is
isolated in CJ(h).
For the converse, assume the spectrum of h is contained in [-1, 1 - c] U
{1 }. Then the spectral projection P associated with the spectral subset {1}
coincides with the orthogonal projection onto the I'-invariant vectors. Since
10(1 - P) :s; 1 - h, one has
21011e - Pe11^2 :::; 2((1-h)e, e)
2
= ~ 2: v(s)(11e11^2 - ar\?T(s)e, e))
sES