1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

1.5. Completely positive maps g

such that llek(a - 'lj!k(a))ekll < e for every a E ~. Let P;t 0 = 1 - PJC 0 and

notice that P;t 0 e1P;t 0 - el is a compact operator; hence llP;t 0 e1P;t 0 ll = 1.

Let (1 E JC~ be a unit vector such that lle1(1 - (1\\ < e. Now, let IC1 be

the finite-dimensional subspace spanned by /Co and {a(1}aEi U {a*(i}aEi·

Reasoning as above, there exists a unit vector (2 E IC[ such that \\e 2 ( 2 -

(2\I < e. Repeating this n times, we get vectors (1, ... , (n and set ~

2: >..i1^2 (k. Then, for every a E ~'we have

we( a)= L AkW(k (a)

~2s L AkWekCk (a)

~s LA.k'!j!k(a)llek(kll^2

~2s '!j!(a) ~s rp(a).

1.5. Completely positive maps

D

Completely positive maps (and their cousins, completely bounded maps) are
the heart and soul of C* -approximation theory. For a positively complete
treatment of these morphisms see [141].

Definitions, examples and Stinespring's Theorem.

Definition 1.5.1. An operator system E is a closed self-adjoint subspace

of a unital C*-algebra A such that lA E E. The n x n matrices over E,

Mn(E), inherit an order structure from Mn(A): an element in Mn(E) is

positive if and only if it is positive in Mn(A). Note that the existence of a

unit guarantees that E is spanned by positive elements.

A map rp from an operator system E to a (not necessarily unital) C*-

algebra B is said to be completely positive if Cfn: Mn(E) ----+ Mn(B), defined

by

Cfn ([ai,j]) = [rp( ai,j)],

is positive (i.e., maps positive matrices to positive matrices) for every n. We

denote by CP(E, B) the set of completely positive maps from an operator

system E into B.

Following well-established precedent, we use c.p. to abbreviate "com-
pletely positive," u.c.p. for "unital completely positive" and c.c.p. for "con-
tractive completely positive."

Example 1.5.2. A *-homomorphism 1f between C* -algebras is c.p. since the

inflations 1fn are also *-homomorphisms (hence preserve positivity). More

generally, a map rp of the form rp(a) = V*7r(a)V for some *-homomorphism