D. Positive De:fi.nite Functions 467Proof. We first prove the assertion for a E CHr) 0lIB(£^2 ) and f E (C~(r) 0
JIB(.€^2 ))*. Since llwa,1llB2 :::;; llallllfll, taking an approximation, we may as-
sume that a= l:sEr .X(s) 0 a(s), where only finitely many a(s) E JIB(.€^2 ) are
nonzero. Then,
Wa,J(cp) = f (L cp(s).X(s) 0 a(s)) = L f(.X(s) 0 a(s)) cp(s)
sEI'
and Wa,f E <C[r] c Q(r). The other case is similar - one approximates f
instead of a. D
Lemma D.8. There is a canonical isometric isomorphism B2(r) = Q(r)*.
Proof. Since Q(r) c B 2 (r), we have a canonical contraction B 2 (r) --+
Q(r). We need to show it is a surjective isometry. Since
ll'PllB2 = sup{lwa,1(cp)I: llall :::;; 1, llfll :::;; 1},
it follows that B 2 (r) --+ Q(r) is an isometry. Let 'I/; E Q(r) be given and
define cp E £^00 (r) by cp(s) = ('I/;, 58 ). Since cp = 'lj; on <C[r], one sees that
cp E B 2 (r) and cp ='I/; on Q(r). D
Lemma D.9. Every w E Q(r) is of the form w = Wa,f for some a E
C~(r) 0 lK(.€^2 ) and f E (L(r) ® JIB(.€^2 ))* (with llall llfll arbitrarily close to
llwllJ.
Proof. Consider the set
S = {wa,1: a E C~(r)0JK(£^2 ), f E (L(r)®JIB(.€^2 ))* with Ila[[:::;; 1, llfll '.S 1}.
Since £^2 EB £^2 ~ £^2 , the set S is a convex subset of the closed unit ball of
Q(r) such that
l['PllB 2 = sup{lwa,J(cp)I : Wa,f ES},
for every cp E B 2 (r). By the Hahn-Banach separation theorem, Sis norm-
dense in the closed unit ball of Q(r). Hence, for every norm-one ele-
ment w E Q(r), there exists a sequence (wan,f,,J~=O in S such that w =
l:~=O 4-nwan.!n· We will view elements in C~(r) 0lK(£^2 0£^2 ) (resp. L(r) ®
JIB(.€^2 0 £^2 )) as oo x oo matrices with entries in C~(r) 0 lK(.€^2 ) (resp. L(r) ®
JIB ( £2)). It follows that
a:= diag(2-nan) E C~(r) 0lK(£^2 0£^2 )
and f E (L(r) ® JIB(.€^2 0 £^2 ))*, defined by
00
n=Osatisfy Wa,f = l:~=O 4-nwan.!n = w. This completes the proof. D