464 D. Positive Definite Functionsunitary representation 7r of r on H such that cp(s) = P'h'.7r(s)l'h'. for every
s E I'. Therefore, we may induce positive definite functions.Lemma D.2. Let r be a discrete group and A ::S r be a subgroup. If
cp: A -+ JE(H) is a positive definite function, then the function
Ind1_ cp: r 3 s f-t L esx,x Q9 cp(O"(sx)-^1 sO"(x)) E JE(£^2 (r/A) 01i)
xEr/A
is positive definite, where O": r I A -+ r is a fixed cross section.Schur multipliers. Let r be an index set. A kernel is a function k: rxr-+
<C. The kernel k is positive definite if for any finite sequence s1, ... 'Sn Er,
the matrix [k(si, sj)]i,j E Mn(<C) is positive. (Sometimes this property is
called positive semidefinite.) When r is a group, a function cp on r is positive
definite if and only if the kernel k(s, t) = cp(s-^1 t) is positive definite. Every
x E JE(£^2 (r)) can be represented as a r x r matrix: x = [xs,t]s,tEr, where
Xs,t = (x8t, 8s) for s, t Er. The Schur multiplier mk: JE(£^2 (r)) -+ JE(£^2 (r))
is defined by mk([xs,tD = [k(s, t)xs,tl· This may not be well-defined because
[k(s, t)xs,t] may not belong to JE(£^2 (r)).
Theorem D.3. Let k: r x r--+ <C be a kernel with k(s, s) = 1 for alls Er.
The fallowing are equivalent:
(1) the kernel k is positive definitei
(2) there exist a Hilbert space 1i and unit vectors ~s E 1i such that
k( s, t) = (~t, ~s) for every s, t E ri
(3) the multiplier mk is a {continuous) u.c.p. map on B(£^2 (r)).
Proof. (1)=?-(2): This follows from a standard GNS construction. (Mimic
the Hilbert space construction following Definition 2.5.6 with a one-variable
positive definite function replaced by k.)
(2)=?-(3): Let V: £^2 (r) -+ £^2 (r) 01i be the isometry such that V 8t =
8t Q9 ~t for t E r. It is not hard to see that mk(x) = V*(x Q9 l)V for
x E JE(£^2 (r)).
(3)=?-(1): For any finite subset E c r, let x E JE(£^2 (r)) be such that Xs,t
is the characteristic function of Ex E. Since xis positive, so is mk(x). This
shows that [k(s, t)]s,tEE is positive for any finite subset E c r and hence k
is positive definite. D
Theorem D.4. Let k: r x r -+ <C be a kernel. The multiplier mk is com-
pletely bounded with llmkllcb ::S 1 if and only if there exist a Hilbert space 1i
and vectors ~s, 'f/t E H, with ll~sll, ll'fltll ::S 1, such that k(s, t) = ('fJt, ~s) for
every s, t Er.