1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.5. Continuity of tensor product maps 83

Proof. It's well known, and easily verified, that the transpose map is a

positive isometry. Let {ei,jh:=:;i,j:=:;n be a system of matrix units for Mn(C)

and consider

n

S := L ei,j ® ej,i·

i,j=l

Evidently S is a permutation matrix - hence unitary - and has norm one.

(If {8k} is an orthonormal basis, then S(8k ® 81) = 81 ® 8k.) On the other

hand n

cp ® idMn(IC) (S) = L ej,i ® ej,i

i,j=l

and a straightforward computation shows that this matrix is equal to nP

where P is the one-dimensional projection onto the span of the vector

D

Unlike the case of states, this shows that the tensor product of norm-one
maps need not have norm one - even on the 2 x 2 matrices when one map
is the identity and the other is a positive unital isometry! The next result
follows easily from the previous one.
Proposition 3.5.2. Let cp: A --> A be a positive, unital isometry. It can
happen that cp 8 idA: A 8 A --> A 8 A is unbounded. For example, let cp be
the transpose map on the unitization of the compact operators.

Not wanting to dwell on the problems that occur for more general maps,
let's treat the case of c.p. maps and move on. We will need the following
result approximately ~o times (maybe more).

Theorem 3.5.3 (Continuity of tensor product maps). Let cp: A--> C and
'ljJ: B--> D be c.p. maps. Then the algebraic tensor product map

cp8'1/J: A8B--> C0D
extends to a c. p. (hence continuous) map on both the minimal and maximal
tensor products. Moreover, letting cp ®rnax 'ljJ: A ®rnax B --> C ®rnax D and
cp ® 'ljJ: A ® B --> C ® D denote the extensions, we have

[[cp ®rnax 'l/J[[ = [[cp ® 'l/J[[ = [[cp[[ [['l/J[[.

Proof. We first handle the spatial tensor product case. Assume CC JB(H)
and D c JB(JC). Let 7rA: A --> JB(H), 1fB: B --> JB(JC) be the Stinespring
dilations of cp and 'lj.J, respectively, and VA: 1i --> H, VB: JC --> fC the as-
sociated bounded linear operators. By Exercise 3.3.4 there is a natural