1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

16 1. Fundamental Facts

There is a similar characterization of c.p. maps from a C*-algebra A into

Mn(q. For a linear map cp: A-+ Mn(C), we define a linear functional cp on

Mn(A) by

n

cp([ai,j]) = L cp(ai,j)i,j·

i,j=l

The notation cp(ai,j)i,j means the (i,j)th entry of the matrix cp(ai,j)· Yes,

the formula is slightly complicated, but it is explicit and very important.

Proposition 1.5.14. Let A be a unital C*-algebra. A map cp: A-+ Mn(C)

is c.p. if and only if cp is positive on Mn(A). In other words,

CP(A, Mn(C)) 3 cp 1----t cp E Mn(A)+

is a bijective correspondence.

Proof. Let { (i}f=l be the standard orthonormal basis for .e; and let ( =

[(1, ... , (nJT E (.e;r. Since

cp([ai,j]) = (cpn([ai,j])(, ()

for [ai,j] E Mn(A), positivity of 'Pn implies that of cp. This proves the "only

if' part. To prove the "if' part, assume cp is positive and let (rr, }-{, e) be the

GNS triplet of cp. Let {ei,j} be the standard matrix units for Mn(C), which

we also view as elements in Mn (A). Then, for the operator V : .e; -+ }-{

defined by V(j = 7r(e 1 ,j)e, it is not hard to check

<p(a) ~ v·"f ·.. J )V:

It follows that cp is c.p. D

Lemma 1.5.15. Let E C A be an operator subsystem and 'ljJ: E -+ C be a

positive linear functional. Then 111/Jll = 'lj;(l). Hence, any norm-preserving

extension of 'ljJ to A is also positive.

Proof. Fix x EE such that llxll :::; 1and11fl(x)I is close to 111/Jll· Multiplying
by a complex scalar of norm one, we may assume that 0 < 'lj;(x) E JR. Since
positive maps are automatically self-adjoint, we have
1
'lj;(x) =
2
1/J(x + x*)

and thus we may assume xis self-adjoint. However, in this case we have the
operator inequality x:::; llxlll and hence

1/J(x) :::; 1/J(l)llxll·

Since a functional satisfying the equation 'lj;(l) = 111/Jll is necessarily positive,
we are done. D