1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

452 B. Operator Spaces

Proof. Without loss of generality, we may assume that A is unital. Since

cp: X ----+ IIB(7t) is c.c., it follows from Theorem B.5 that S'P: Sx ----+ M2(IIB(7t))

is u.c.p. Let (7r, it, V) be a Stinespring triplet of a u.c.p. extension of S'P

to M2(A). We observe that it can be decomposed.as it= ii E9 ii in such

a way that IIB(it) = M2(IIB(ii)) and if: M2(A) ----+ M2(IIB(ii)) is of the form

id2 ® 1f, for some *-representation 1f : A ----+ JIB ( 1-l). Under this identification,

V = [Vi,j] E M2(IIB(7t, ii)). But since

V * [1 0 0 OJ V = [1 0 0 OJ and V * [o 0 OJ 1 V = [o 0 OJ 1 '

V := Vi,1 and W := Vz,2 are isometries from 1t into ii (and Vi,2 = 0 = Vz,1).

Hence,

V*7r(x)W = [1 OJ [6 ~r rn 1f~x)J [6 ~] [~]

=[1 OJS'P(rn ~])[~]

= cp(x)

for every x EA. D

Let X C A and Y c B be operator spaces. We define the minimal

tensor product X ® Y of X and Y to be the norm closure of X 8 Y in A® B.

The proof of the following corollary is very similar to C* -results in the

body of this text (cf. Theorem 3.5.3 and the proof of Proposition 3.3.11).

Corollary B.8. Let Xi c Ai and Yi c Bi be operator spaces (i = 1, 2) and

let cp: X 1 ----+ X 2 and 'ljJ : Y1 ----+ Y2 be c. c. maps. Then,

cp ® 'l/J: X1 ® Yi ----+ X2 ® Y2

is a c.c. map. For z EX® Y, one has

/[z/[rnin =sup //(cp ® 'lfJ)(z)l/Mrn(<C)®Mn(<C)

where the supremum is taken over all m, n E N and c. c. maps cp: X ----+

Mm(CC) and 'l/J: Y----+ Mn(CC).

Though we won't prove it, we remind you that for C*-algebras A, B, C
and a c.c. map cp: A ----+ B, the map cp ® iclc: A 8 C ----+ B 8 C need not be
continuous with respect to the maximal tensor norm ([88]).
It follows from Theorem B.7 that if cp is unital and c.c., then V =Wand
cp is automatically c.p. More generally, we have the following perturbation
fact: