1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

128 4. Constructions

Using the inequality (a - b)^2 :::; Ja^2 - b^2 J for all positive numbers a, b, we

then get

lls *a Ti -Till~= sup (2:: h/s.mr

1

·x(g) - Jmf(g)l2)

xEX gEI'

:::; sup (2:: Js.mr

1

·x(g)-mf(g)J)

xEX gEI'

x~.y sup (L Js.mf(g) - mt'Y(g)J)

yEX gEI'

=sup [[s.mf - mt"Yll1-+ 0.

yEX

Hence the Ti's have the right properties, except for finite support. Fixing

this problem is easy once we prove the following claim.

Claim. If T: I'-+ C(X) is a positive function such that (T, T) = lC(x),

then there exists a sequence of finitely supported positive functions Tn: r -+

C(X) such that (Tn, Tn) = lc(x) for all n and

J[s *a Tn - Tnll2-+ JJs *a T-TIJ2,

for alls EI'.

To prove this claim, we let Fn C Fn+l be a sequence of finite subsets of
r such that LJ Fn =I'. Since

L T(g )^2 = lc(x)

gEI'

and convergence is uniform, it follows that for all sufficiently large n,

L T(g)2 > 0,

gEFn

meaning bounded uniformly away from 0. Hence we can define Tn by declar-
ing
1
L gEF.,,, T( g )2T(g),

for all g E Fn and Tn(g) = 0 if g tJ. Fn. Tedious and unenlightening calcula-
tions (left to the diligent few) show that these functions do the trick..

To prove the opposite direction of Lemma 4.3.7, one basically reverses
the procedure above. That is, define mf(g) = Ti(g)^2 (x) and calculate away.
It should be noted that the Cauchy-Schwarz inequality gets used in the
following way:

L Jar - brl = L lai - bil(ai +bi):::; ll(ai)-(bi)ll2ll(ai) + (bi)ll2-

D