1.5. Completely positive maps 13(2) E is c.c.p.;
(3) E is contractive.Proof. We only have to prove that the last condition implies the first, so
assume E is contractive. Passing to double duals, we may assume that A
and B are von Neumann algebras. We first prove that E is a B-bimodule
map. Since von Neumann algebras are the (norm) closed linear span of their
projections, it suffices to check the module property on projections. Let
p EB be a projection and let pJ_ = lA - p. Since pE(pJ_x) = E(pE(pJ_x))
for every x E A, we have that for any t E ffi.,
(1 + t)^2 llpE(pJ_x)ll^2 = llpE(pJ_x + tpE(pJ_x))ll^2
:::; llPJ_X + tpE(pj_x) 112
:::; llpj_xll^2 + t^2 llpE(pJ_x)ll^2 •
It follows that llpE(pJ_x) 112 + 2tllPE(pJ_x) 112 :::; llpJ_xll^2 for all t E ffi.; hence,
pE(pJ_x) = 0. In particular, E(l~x) = lBE(l~x) = 0. The same reasoning
shows (lB - p)E(px) = 0. It follows that
E(px) = pE(px) = pE(x - pJ_x) = pE(x)
for every projection p E B and x E A. Switching to the other side, one
shows E(xp) = E(x)p as well - hence Eis a B-bimodule map.
Since E is unital - indeed, bE(lA) = E(b) = b for any b E B - and
contractive, E is necessarily positive (this follows from the corresponding
fact for functionals). To prove that E is c.p., let a positive element [xi,j] E
Mn(A) be given. Let ?r: B--+ IBl(7-i) be any *-representation with a cyclic
vector e. Then, for any bi, ... , bn E B, we have
L (7r(E(xi,j) )7r(bj )e, 7r(bi)e) = (7r(E(L b';xi,jbj) )e, e) 2:: 0
i,j i,jsince l:i,j b';xi,jbj 2:: 0 in A. It follows that [7r(E(xi,j))]i,j 2:: 0 in Mn(7r(B)).
Since 1f is an arbitrary cyclic representation, we conclude [E(xi,j)]i,j 2:: 0 in
Mn(B). D
The following basic fact is extremely useful.Lemma 1.5.11. Let M be a van Neumann algebra with a faithful normal
tracial state T and let lM ENC M be a van Neumann subalgebra. Then,
there exists a unique trace-preserving, normal conditional expectation E from
M onto N.
Proof. The restriction, of T to N will also be denoted by T. Let a, y E M
be arbitrary and a= ulal be the polar decomposition of a. We claim that
IT(ya)I = IT(yulal)I:::; T(yulaluy)^112 T(lal)^112 :::; llvllT(lal).