450 B. Operator Spaces
::::; ll'Pll sup{ll L~i1JjXi,jll: L l~il^2 =1=L117jl
2
}
i,j i j
::::; ll'Pllll[xi,j]ll·If X c A is an operator system and <p: X ~ B is c.p., then <p is c.b.
and ll'Pllcb = ll'Pll = ll<p(l)ll· Indeed, this follows from Arveson's Extension
Theorem and Stinespring's Dilation Theorem.
Remark B.3. If <p: X ~ Y is a c.b. map, then it is not hard to see that
for any infinite-dimensional Hilbert space H,
ll'Pllcb = ll<p®idlIB(?-i'.)11·Indeed, the inequality ::::; is immediate by compressing to finite-dimensional
subspaces. For the other inequality, one shows that for T E X 8 IIB(H),
ll'P ® idlIB(?-l)(T)ll is the supremum over compressions to finite-dimensional
subspaces of H.
The following fact can be quite useful.
Lemma BA (Smith's lemma). Let X be an operator space and <p: X ~
Mn(C) be a bounded linear map. Then,
ll'Pllcb = IJidMn(C) ® 'PJI.Proof. Since ll'Pllcb = IJidJIB(£2) ® <pJI by Remark B.3, for any c > 0, there is
a contraction x E IIB(.€^2 ) ® X and unit vectors~' 17 E .€^2 ® .e; such that
J((idlIB(£2) ® <p)(x)17, ~)J > JJ'Pllcb - €.
Note that there are n-dimensional Hilbert subspaces H, JC, c .€^2 such that
~ E 1t ® .e; and 1J E K ® .e;. It follows that JlidlIB(JC,?-l) ® 'Pll > ll'Pllcb - c.
Since IIB(JC, H) is spatially isomorphic to Mn(C) (i.e., there are unitaries
U: IC, ~ .e; and V: 1-{ ~ .e; such that Mn(<C) ~ VIIB(K, H)U* completely
isometrically) and c > 0 was arbitrary, we are done. D
Many facts about c.p. maps can be transferred to results about c.b. maps
via an indispensable trick of Paulsen. For an operator space X in a unital
C*-algebra A, we define an operator system Bx c :Mb(A) by
Sx = {[)..:*A μ~A]:>..,μ E cc, x,y EX}.
For a map <p from X into a unital C* -algebra B, we define a map Sep: Sx ~
M2(B) by
S ( [ Al A x ] ) = [ AlB <p( x) ]
ep y* μlA <p(y)* μlB.Theorem B.5 (Paulsen). The map Sep: Sx ~ M2(B) is u.c.p. if and only
if the map <p : X ~ B is c. c.