82 3. Tensor ProductsThus, for a unit vector~ E eJH, we have ll(a@b)(~) -~II:::; 2c:. Since E
was arbitrary, Ila Q9 bll = 1. 0
Another proof of Proposition 3.4. 7. We must show c.p Q9 1/J is continu-
ous on A @a B. By convexity arguments, we may assume that c.p and 1/J are
pure. Find some excising nets ( ei) and (Ji) for c.p and 1/J, respectively (see
Theorem 1.4.10). It follows that for x = L::~=l ak Q9 bk EA 8 B, we have
llxlla 2:: lim i II ( ei@ fi)x( ei@ fi) II = lii;n i II ( c.p@ 1/J) (x )( ei@ fi)^2 II = I ( c.p@ 1/J )(x) I·
0Exercises
Exercise 3.4.1. Let 7f: A@B---+ C be a *-homomorphism which is injective
when restricted to A 8 B. Show that 7f must be injective on all of A Q9 B.
Is this still true if one replaces II · II min by II · II max?
Exercise 3.4.2. Prove that A @a Bis simple if and only if II· Ila= II· llmin
and both A and B are simple. (Hint: For the "if' direction it suffices
to show that every irreducible representation of A Q9 B is faithful. But if
7r: A Q9 B ---+ .IIB(H) is irreducible, then both 1fA(A)" and 1fB(B)" must be
factors. One then uses the previous exercise and a theorem of Murray and
von Neumann which asserts that if MC .IIB(H) is a factor, then the product
map
M 8 M' ---+ .IIB(H)is injective (see [183, Proposition IV.4.20]).
3.5. Continuity of tensor product maps
Given how delicate the proof of Takesaki's Theorem is, it's no surprise that
continuity of maps on tensor products requires some care. It turns out that
nothing funny happens so long as one sticks to c.p. maps, but this is the
largest class of maps which always behave well. To get a feel for what can
go wrong, let's consider a finite-dimensional example.
Proposition 3.5.1. Let c.p: Mn(CC) ---+ Mn(CC) be the usual transpose map
on the n x n matrices. Then c.p is a unital, positive isometry but the norm
of
!.p@ idMn(IC): Mn(CC)@ Mn(CC) ---+ Mn(CC) @Mn(CC)is greater than or equal to n.^13
(^13) Actually, it is equal to n, but we won't need this fact.