1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

206 5. Exact Groups and Related Topics

We note that sup1'EG lh(1)I = supxEG(o) h(x) for a positive-type function

h. Also, h E Cc( G) is of positive type if and only if >..(h) is positive in C~ ( G).

Proposition 5.6.16. Let h: G -+ <C be a continuous positive-type function

with sup lh(1)I ~ 1. Then, the multiplier map

mh: Cc(G) 3 f 1-> hf E Cc(G)

extends to a c.c.p. map on C~(G) and on C*(G).

Proof. We first deal with C~ ( G). For every x E G(o), let mh, be the Schur

multiplier of [h(a,8-^1 )]a,,BEGx on IB\(.€^2 (Gx)). Since his of positive type, mh,

is a c.c.p. map (Theorem D.3). For every f E Cc(G) and x E G(o), we have

Ax(hf) = mh,(>..x(J)). Since {Ax}xEG(o) is a faithful family, we are done.

We give an ad hqp proof for C* ( G) assuming that h = (* * ( for some

( E Cc(G) with llCllL2(G) ~ 1. (This case is good enough for.these notes.

The general case requires representation theory of groupoids [140].) Take a

partition of unity {Pi} C Cc(G) such that the source map is homeomorphic

on SUPP Pi· We define (i E Cc(G(O)) by the formula (i(s(r)) = ((1)Pi(r) so

that ((r) = l:i (i(s(r)). We may assume that all but finitely many (i are

identically zero. Observe that

An element in Mn(Cc(G)) is called algebraically positive if it is a finite sum

of matrices of the form [g; * gq]p,q. (By Lemma 4.2.1, any positive element in

Mn ( C* ( G)) is approximated by algebraically positive elements.) We claim

that mh is completely algebraically positive - i.e., id© mh: Mn(Cc(G))-+

Mn (Cc ( G)) maps algebraically positive elements to algebraically positive

elements, for all n E N. Let f E Cc(G) be given and define j E Cc(G) by

J = l: fi, where fi(I) = f(r)(i(s(r)). Then,

(f* * f)(1) = L I: fi(,e)fj(,81)

i,j ,BEGr("Y)

= L L f(,8)(i(r(1))f(,81)(j(s(1))

i,j ,BEGr("Y)

= h(r) L f(,B)f(,81) = mhU* * f)(r).

,BEGr("Y)

It follows that mh is algebraically positive. The proof of complete algebraic
positivity is similar. We observe that for any f E Cc(G(O)) with llflloo ~ 1
and any 91, ... ,gn E Cc(G), the element

[g; 9q]p,q - [g; f f 9q]p,q = [.9; gq]p,q E Mn(Cc(G))