1.5. Completely positive maps 11Remarks 1.5.5. We call the triplet (1f, it, V) in Theorem 1.5.3 a Stinespring
dilation of rp. When r.p is unital, VV = rp(l) = 1 and hence V is an
isometry in this case. The projection VV E JB(it) is called the Stinespring
projection. In general there could be many different Stinespring dilations,
but we may always assume that a dilation ( 1f, it, V) is minimal in the sense
that 1f(A)V'H is dense in it (which holds for the construction used in the
proof above). Under this minimality condition, a Stinespring dilation is
unique up to unitary equivalence ..
When we come to c.p. maps and maximal tensor products, the following
result will be crucial (see Theorem 3.5.3): If (1f, it, V) is a minimal Stine-
spring dilation of r.p : A ---+ lB ( 'H), then the commutant r.p (A)' c lB ( 'H) also
lifts to JB(it).
Proposition 1.5.6. Let (1f, it, V) be the minimal Stinespring dilation of a
c.c.p. map r.p: A---+ JB('H). Then, there exists a -homomorphism
p: r.p(A)'---+ 7r(A)' c JB(it)
such that
r.p(a)x = V7r(a)p(x)V
for every a E A and x E rp(A)'.
Proof. For x E rp(A)', we define a linear operator p(x) on the span of
7r(A)V'H by
p(x) ( ~ 1f(ai)vei) = ~ 1f(ai)vxei·
i i
Once we prove that p(x) is well-defined and bounded for every x E rp(A)', it
is not too hard to check that p gives rise to a -representation of r.p(A)' on
it such that p(r.p(A)') C 1f(A)' and r.p(a)x = V1f(a)p(x)V for every a E A
and x E rp(A)'.
So, let x E rp(A)' and I:i 1f(ai)vei E 7r(A)V'H be given. If we set
e = [6, ... 'enV E 'Hn and let diag(x) denote the n x n matrix with x's
down the diagonal and zeroes elsewhere, then we have
llp(x) I: 1f(ai)vei\lit - L:(xr.p(aiaj)xej,ei)?-l
i,j
(diag(x ) r.pn ([ai aj]) diag(x )e' e)'J-ln
< \lxll^2 (rpn([aiaj])e, e)'J-ln
llx\1^2 11I:1f(ai)veillit,
where we use the fact that diag(x) and r.pn([aiaj]) E Mn(rp(A)) commute in
the third line above. Therefore, we have llp(x)ll :::; llxlJ as desired. 0