84 3. Tensor Products
-):iomomorphism 1fA ® 1fB: A® B ------ llll(H ® K). Hence we may define
cp ® 'ljJ: A® B------ C ® D by the formula
cp ® 'ljJ(x) =(VA® VB)7rA ® 1fB(x)(VA ®VB)·
Note that on elementary tensors we have
cp ® 'l/J(a ® b) = (VA7rA(a)VA) ® (V_B7rB(b)VB) = cp(a) ® 'l/J(b).Hence cp®'l/J really is a c.p. extension of cp0'1/J which takes values in C@D C
llll(H ® K). Finally note that the completely bounded norm (cf. Appendix
B) satisfies
JJcp ® 'l/JJ[cb ~ JJVA ® VBJJ^2 = JJVAJl^2 JJVBJl^2 = J[cpJJJJ'l/JJJ.The other inequality is easy and will be left to the reader (consider elemen-
tary tensors and Lemma 3.4.10).
For the maximal tensor product, let's first tackle the case that B = D
and 'ljJ = idB. Fix a faithful representation C ®max B C IIB(H). By the
existence of restrictions, we may assume that C c IIB(H) and B C llll(H)
commute (and generate C ®max B) thus allowing us to regard cp as a c.p.
map into llll(7-i) with B C cp(A)'. Applying Stinespring to cp - also lifting B
with the commutant cp(A)' (Proposition 1.5.6) - we get a -representation
of A ®max B (by universality) which we can cut to recover the original map
cp 0 idB: A 0 B------ C ®max B c llll(H) (just as in the spatial tensor product
case above).
Since an arbitrary map cp 0 'ljJ: A 0 B ------* C 0 D can be decomposed as
( cp 0 idn) o (idA 0 'ljJ), the proof is complete. D
Remark 3.5.4. It is not hard to extend the previous result to minimal
tensor products and c.b. maps. The corresponding result for maximal tensor
products is not true ([88]).
The previous theorem extends to operator spaces (see Remark B.3; note
that any operator space sits inside llll(H)).
Corollary 3.5.5. If cp: W ------* Y and 'ljJ: X ------* Z are c. b. maps (resp. u. c.p.
maps) of operator spaces (resp. operator systems), then the tensor product
map cp 0 'ljJ extends uniquely to a c.b. map (resp. u.c.p. map)
cp ® 'ljJ: w ® x ------* y ® z.
Moreover, JJcp ® 'l/Jllcb = JJcpJJcbJJ'l/JJlcb·Since C*-norms are always cross norms (Lemma 3.4.10) and we can al-
ways extend c.p. maps to the minimal and maximal tensor products, the
next result is a triviality (which we will use frequently and without refer-
ence).