1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

454 B. Operator Spaces

Proof. Let B c IIB(H). Then, Corollary B.9 provides a u.c.p. map 1/J': E----*
IIB(H) with ll'P -1/J'llcb :S 2(ll'Pllcb - 1). Let O" = 1/J' - <p and n = dim(E).
We claim that there is a positive linear functional fJ on E, with llOll :S nllO"ll,
such that fJ - O" is c.p. If this is the case, then defining 1/J = <p + fJ, we are
done. Fix a basis {xi}r=l for E as in Lemma B.10 and let fJ = llO"ll i:~=l lxil·
For every a 2: 0, we have

n n n

O"(a) = Lxi(a)O"(xi) :SL lxil(a)llO"(xi)ll :S 110"11 L IXil(a) = fJ(a)

i=l i=l i=l

and hence fJ - O" is positive. The proof of complete positivity is similar. D

We close this section with a c.b. variant of Corollary 1.6.3, though we

leave the proof to the reader. (Hint: Theorem B.5.)

Lemma B.12. Let X C IIB(H) be an ultraweakly closed operator space and

let <p: X ----* Mn ( (['.) be a c. c. map. Then, there exists a net of ultra weakly

continuous c.c. maps <p>..: X ----* Mn(C) which converges to <p in the point-

norm topology.

Operator space duality. Let X be an operator subspace of a C* -algebra

A. In many cases, the position of X inside A does not matter. What matters

is the operator space structure of X - i.e., the norms on Mn(X), n EN, in-

herited from Mn(X) c Mn(A). Hence, we often identify the operator space

X with another operator space which is completely isometrically isomor-

phic. In other words, we consider an operator space as an abstract Banach

(or normed) space equipped with a distinguished family of matrix norms

on Mn(X) (satisfying some axioms) and view X CA as a concrete realiza-

tion of X. This viewpoint is very similar to that for C* -algebras and their

representations. Like the Gelfand-N aimark Theorem for C* -algebras, there

is Ruan's Theorem which gives an axiomatic characterization of operator

spaces, but we won't need it (we just had to mention it - see [63]).

Recall that for operator spaces X C A and Y c B, we have X ® Y C

A ® B and the operator space structure of X ® Y depends only on those of

X and Y (cf. Corollary B.8).

There is a very important notion of dual operator spaces which was

introduced in [21, 22, 62]. Let X C IIB(H) be an operator space and let X*

be its dual Banach space. For x = [xk,Z] E Mm(X), we define Ox: X* 3 <p 1-+

[<p(xk,z)] E Mm((['.). Denote by Ballm(X) the closed unit ball of Mm(X).

One puts an operator space structure on X* by the isometric inclusion

II

mEN xEBallm(X)