Positive Definite
Functions, Cocycles
and Schoenberg's
Theorem
Appendix D
Unitary representations. As in Section 2.5, r denotes a discrete group,
and C[r] is its complex group algebra. Let (rr, H) and (7r', H') be two unitary
representations of r. Recall that 7r and 7r^1 are unitarily equivalent if there
exists a unitary U : 1{ --+ H' such that 7r^1 ( s) = U 7r ( s) U for every s E
r. We say 1f^1 is contained in 7r if 7r^1 is equivalent to a subrepresentation
of 7r; 1f^1 is weakly contained in 7r if the map 7r(s) r--+ 7r^1 (s) extends to a
(continuous) -homomorphism from C(7r(r)) into C(7r'(r)); 7r and 7r^1 are
weakly equivalent if the map 7r ( s) r--+ 1f^1 ( s) extends to a -isomorphism from
C(7r(r)) onto C*(7r'(r)).
Generalizing Definition 2.5.6, positive definite functions make perfectly
good sense for operator-valued functions.
Definition D.1. We say a function cp: r --+ JB(H) is positive definite if
for any finite sequence s1, ... , Sn E r, the operator matrix [cp(si^1 sj)]i,j E
Mn(lB(H)) is positive.
One should check that Theorem 2.5.11 holds mutatis mutandis for an
operator-valued function. In particular, cp: r --+ JB(H) with cp( e) = 1 is
positive definite if and only if there exist a Hilbert space H =:i 1{ and a
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