466 D. Positive Defi.nite Functions
Let r be a discrete group and cp: r ---+ CC be a function. Abusing notation,
we denote by m'P the Schur multiplier on JIB( £^2 (r)) that is associated with the
kernel (s, t) t--t cp(sr^1 ). This notation is compatible with Definition 2.5.10:
m'P(>.(s)) = cp(s)>.(s) for every s Er.
Proposition D.6. Let r be a discrete group and cp: r ---+ CC be a function.
The multiplier m'P on JIB( £^2 (r)) is completely bounded if and only if it is
completely bounded on C{(r). Moreover, their cb-norms coincide.
Proof. Suppose that m'P is completely contractive on C{ (r). Let U be the
unitary operator on £^2 (r) @£^2 (r) given by U5 8 0 5t = 58 0 58 t for s, t Er.
An easy computation shows
for every s, t E r. It follows that
U(m'P(x) 0 l)U = (idlIB(£2(r)) 0 (m'P)Jc~(r))(U(x 0 l)U)
for any x E OC(£^2 (r)). Hence, m'P is completely contractive on OC(£^2 (r)).
By ultraweak continuity, this implies that m'P is completely contractive on
JIB(£^2 (r)). D
We note that the norm of m'P on C{ (r) does not coincide with the cb-
norm in general.
A function cp on a group r is called a Herz-Schur multiplier if m'P is
completely bounded. (Sometimes one also refers to m'P as a Herz-Schur
multiplier.) We denote by B 2 (r) the Banach space of Herz-Schur multipli-
ers equipped with the Herz-Schur norm defined by //cp//B 2 = l/m'Pl/cb· Let
cp: r ---+ CC be a function and w'P be the corresponding linear functional on
qr] (cf. Definition 2.5.10). If w'P is bounded on C*(r), then cp is a Herz-
Schur multiplier with /lcpl/B 2 ~ /lw// (cf. proofofTheorem2.5.ll, (3)::::;,. (4)).
In particular, //cp/IB 2 ~ l/cp/1 2 for every cp E £^2 (r). (This fact also follows
easily from Theorem D.4.) It follows that the norm closure of CC[r] in B 2 (r)
contains £^2 (r).
A finitely supported function w E CC[r] defines an element of B 2 (r) by
the formula w(cp) = I:sErw(s)cp(s). We denote by Q(r) the norm closure
of CC[r] in B 2 (r). For a E C{(r) @JIB(£^2 ) and f E (C{(r) @JB(£^2 )) or for
a E L(r) ® JIB(£^2 ) and f E (L(r) ® JIB(£^2 )), we define Wa,J E B 2 (r)* by
Wa,J(cp) = f(m'P 0 id( a)).
It is clear that /lwa,J/IB:i ~/la/II/fl/.
Lemma D. 7. For every a and f as above, Wa,f E Q(r).