B. Operator Spaces 453Corollary B.9. Let E C A be an operator system and <p: E ~ JEB(7i) be a
unital self-adjoint map (i.e., <p(l) = 1 and <p(a*)* = <p(a) for a EE). Then,
there exists a u.c.p. map 'ljJ: E ~ JEB(7i) with ll'P - 'l/Jllcb::::; 2(ll'Pllcb - 1).Proof. Let .\. = ll'Pllcb· By Theorem B.7, there exist a *-representation
A~ JEB(H) and isometries V, W: 1-{ ~ H such that
<p(a) = .\.V*?T(a)W = .\.W*?T(a)V
for a E E. Let 'ljJ: E ~ JEB(7i) be the u.c.p. map defined by
1
'lj;(a) =
2(V*?T(a)V + W*?T(a)W)
for a EE. Then,
1
.\.'lj;(a) - <p(a) =
2
>..(V - W)*?T(a)(V - W).
Since <p is unital, we have .\. V*W = 1 andll'P -1/Jllcb::::; t.All(V - W)*(V - W)ll + (.\.-1) = 2(.\.-1) ..
DLet E be an n-dimensional linear space with a basis {Xi }r=l · Then, its
dual basis, denoted by {xi}f= 1 c E*, is defined by the relation xi(xj) = bi,j·
Lemma B.10. Let E be an n-dimensional operator system. Then, there
exists a basis { Xi}f=l consisting of self-adjoint elements such that llxi II
1 = llxill for every i.
Proof. Fix a basis {zi}f= 1 for E and consider the multilinear function
: En 3 (y1, ... ,yn) ~ det[:i;;(yj)] EC.
We denote by Ball(E)sa the self-adjoint part of the. closed unit ball of E.
Since Ball(E)~a is compact, there exists (x1, ... , Xn) E Ball(E)~a which at-
tains the maximum of the absolute value of on Ball(E)~a· It is not hard
to see that { Xj }j= 1 is a basis for E and {xi}f= 1 CE* given by.
Xi(Y) = (x1, ... , Xn)'-^1 (x1, ... , Xi-1, y, Xi+l, ... , Xn)
forms the dual basis. Since Xi is self-adjoint, the maximality assumption on
{xj}j= 1 yields that 11~1.1=1 for every i. D
The following is a variant of Corollary B.9, which works for maps into a
unital C*-algebra B rather than into JEB(7i).
Corollary B.11. Let E be a finite-dimensional operator system, B be a
unital C* -algebra and <p: E ~ B be unital self-adjoint map. Then, there
exists a u.c.p. map 'lj;: E ~ B with ll'P-'l/Jllcb::::; 2dim(E)(ll'Pllcb -1).