1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

B. Operator Spaces 451

For the proof, we need the following lemma.

Lemma B.6. For any operators a, b;:::: 0 and x, we have

[ :* ~ ] ;:::: 0 <====} Ve> 0 ll(a + c1)-^1 l^2 x(b + d)-^11211 ~ 1.

Proof. Let z(e) = (a+ci)-^112 x(b+c1)-^112. The follow facts all make good

exercises:

[ x* a x b ] > -^0 <====} Ve >^0 [

a+d x ]>o

x* b+ cl -

<====} Ve> O [^1 z(e) ] > 0

z(e)* 1 -

<====} Ve> (^0) llz(e)ll ~ 1.
D
Proof of Theorem B.5. The 'only if' part is trivial. To prove the other
direction, let T = [Ti,j] E Mn(Sx) be a given positive element, where
Ti,j = [ ,\:,j x~,~ ] E Bx c M2(A).
Yi,j μi,J
We canonically identify Mn(M2(A)) with M2(Mn(A)) and Mn(M2(B)) with
M2(Mn(B)). Let ,\ = [Ai,j], μ = [μi,j], x = [xi,j] and y = [Yi,j]· Under the
above identification, T is of the form
T = [ ,\ x ] E M2(Mn(A)).
Y μ
Since Tis positive, ,\andμ are positive and y = x. It follows from Lemma
B.6 that II(.+ c1)-^112 x(μ + e)-^112 11~1 for any e > 0. Since cp is c.c.,
II(.+ el)-^112 cpn(x)(μ + d)-^11211 = ll'Pn((,\ + d)-^112 x(μ + d)-^112 )11~1
for any e > 0. This means
(Sc,o)n(T) = [ 'Pn~Y) 'Pn~x) ] E M2(Mn(B))
is positive. Hence, Sc,o is c.p. D
Here is an important generalization of Stinespring's Theorem.
Theorem B.7 (Haagerup, Paulsen, Wittstock). Let X CA be an operator
space and cp : X --+ JIB ( H) be a c. c. map. Then, there exist a Hilbert space ii,
a -representation 'Tr: A --+ IIB(ii) and isometries V, W: H --+ii such that
cp(x) = V'Tr(x)W
for every x E X. In particular, cp extends to a c. c. map on A.