D. Positive Definite Functions 465Proof. First suppose that llmkllcb ::S 1. By Wittstock's factorization theo-
rem for completely bounded maps (Theorem B. 7), there exist a Hilbert space
H, a -representation 7r: IBS(.£^2 (f))--+ IBS(H) and isometries V, W: .£^2 (f)--+ 1{
such that mk(x) = VK(x)W for every x E JBS(.£^2 (f)). Fix r E r and set
es= 7r(er,s)V8s and 'T/t = 7r(er,t)W8t. It follows that
('T/t, es) = (V7r(es,t)W8t, 8s) = mk(es,t)s,t = k(s, t).
Conversely, if k(s, t) = ('T/t, es) for some es, 'T/t E 1i with lies ii, ilrttll :'.S 1,
then we define contractions v, w: .£^2 (f) --+ .£^2 (f) ® 1{ by v 8s = 8s ®es and
W8t = 8t ® 'T/t· Now check that mk(x) = V(x ® l)W for x E JBS(.£^2 (f)). D
If k: f1 x f1 --+ C is a kernel and j: f2 --+ f1 is a map, then for the
kernel k1(s, t) = k(j(s), J(t)) on f2, we have llmk 1 llcb ::S llmkllcb·
It turns out that a bounded linear map e: JBS(.£^2 (f))--+ IBS(.£^2 (f)) is a Schur
multiplier if and only if it is an .£^00 (f)-module map - i.e., B(f xg) = JB(x)g
for f, g E .£^00 (r) and x E JBS ( .£^2 (r)). Since .£^00 (r) has a cyclic vector (when r
is countable), any bounded Schur multiplier e is automatically completely
bounded and llBllcb = llBll· The result below comes from [178], inspired by
work of Christensen and Haagerup.
Proposition D.5. Let A c JBS(H) be a C* -subalgebra with a cyclic vector.
Let X c JBS(H) be an operator space such that AX A c X. If cp: X --+ IBS(H)
is a bounded A-A-bimodule map, then it is c.b. with llcpllcb = ll'Pll ·
Proof. The proof is by contradiction. Suppose cp is contractive, but there
aren EN and [Xij] E Mn(X) with ll[xij]ll = 1 such that ll[cp(Xij)]ll > 1. Then
there are c > 0, 6, ... ,en,'T/1, ... ,'f/n E 1i such that L::lleill^2 = L::llrtjll^2 =
1 - c and I l:(cp(Xij)'T/j,ei)I > 1. Let (be an A-cyclic unit vector. We
may assume that ei = ai( and 'T/j = bj( For a = c + 2: aiai, we have
lla^112 (ll^2 = c +I:: lleill^2 = 1; similarly for b = c + 2: bjbj. Let ci = aia-^1!^2
and dj = bjb-^112. We have L::cici ::S 1 and l:djdj ::S 1. It follows that
I L(cp(Xij)'T/j, ei)I =I L(cp(Xij)djb^1 i^2 (, Cia^1 l^2 ()I
i,j i,ji,ji,j
::S II 2::cicill^112 il[xij]l1Mn(X) II Ldjdjll^112
i j
:'.S ll[Xij]llMn(X) = 1.
This is a contradiction. D