436 17. K-HomologyProof. Taking adjoints and replacing a with a, our intertwining assump-
tion also implies
n(a)T = TO"(a).
Now observe that TT commutes with n(A):
TTn(a) = T(O"(a)T) = (TO"(a))T = n(a)TT.
Since n is irreducible, the commutant of n(A) consists of scalar multiples of
the identity and hence TT = .\17-l for some A 2 0. Replacing T by )xT,
we may assume that T is an isometry. Since T is a unitary operator onto
its range, we will be done once we know that the range of T is an invariant
subsp~ce for O"(A) -i.e., TT commutes with O"(A). But this is trivial. D
One often sees property (T) defined as follows: If a unitary representa-
tion weakly contains the trivial representation, then it has nonzero fixed vec-
tors (i.e., actually contains the trivial representation). This one-dimensional
property implies a stronger version of itself.
Proposition 17.2.2. Assumer has property (T), n: C*(I')-+ Mn(C) is
an irreducible representation and O": C* (r) -+ JB(JC) is any representation
such that there exist isometries Vk: .e;;_-+ JC with llO"(g)Vk-Vkn(g)ll-+ 0 for
all g E r.^4 Then 7r is unitarily equivalent to a subrepresentation of O".Proof. Let S 2 (.e;;,, JC) denote the Hilbert space of Hilbert-Schmidt class op-
erators from .e;;, to JC.^5 We define a unitary representation p of r on S 2 (£;;,, JC)
by
p(g): T r-+ O"(g )Tn(g )*.
By Schur's Lemma, it suffices to show that this representation has a nonzero
fixed vector (since this vector will be an intertwiner). Since r has property
(T), we only need to observe the existence of a sequence of asymptotically
invariant unit vectors. A routine calculation shows that the vectors Jn Vk E
S2(.e;;,, JC) have this property. DThough the result above is completely elementary, it has a striking
consequence: The central projection in C* (I')** corresponding to a finite-
dimensional irreducible representation actually lives in C* (r). In this con-
text, finite-dimensional central covers are often called something else.(^4) This is equivalent to requiring llV,;"u(g)Vk -7r(g)jj---+ 0 for all g Er.
(^5) Recall that if S, T E JIB( e;;_, K), then their inner product is defined by
n
(S, T)s 2 = L(Sei, Tei),
i=l
where {ei} is your favorite orthonormal basis of e;;,.