3.4. Takesaki 's Theorem 81Proof. Assume first that both A and B are unital and separable. Then
we can find faithful states c.p E S(A) and 1/J E S(B). Let II · Ila be any
C* -norm on A 8 B and by the previous lemma we may extend c.p 8 'ljJ to a
state c.p ®a 1/J on A ®a B. By uniqueness of GNS representations it follows
that the representations 1fep®"''¢1A 0 B and ?rep 8 1f'¢ are unitarily equivalent.
Since we chose faithful states, both ?rep and 1f'¢ are faithful representations.
Proposition 3.3.11 then implies that the norm closure of ?rep 8 7r'¢(A 8 B) is
isomorphic to A® B. Hence A® B is also isomorphic to 1fep®"''¢(A ®a B)
which, being a quotient of A ®a B, shows that [[ · [[min:::; [[ · Ila·
The nonunital case can be deduced from the unital one with the help
of Corollary 3.3.12. In the nonseparable setting we first fix an arbitrary
element
n
x = Lai ® bi E A 8 B C A ®a B.
i=i
Letting Ai = C(ai, ... , an) and B1 = C(b1, ... , bn), we have a natural
inclusion Ai® B1 CA® B. (Just think about what this should mean and
it will become an obvious consequence of Proposition 3.3.11 -otherwise see
Proposition 3.6.1.) Thus the separable case implies that [[x[[min :::; [[xl[a and
we are done. 0
The following corollary gets used, both explicitly and implicitly, all of
the time. For example, in the literature it is often written that A 8 B has
a unique C* -norm if and only if A ®max B = A® B.
Corollary 3.4.9. For any A and B and any C* -norm [[ · [[a on A 8 B we
have natural surjective *-homomorphisms
A ®max B--+ A ®a B--+ A® B.It is a fact that every C*-norm on AGE is a cross norm (i.e., [[a®b[[a =
[[a[[ [[b[J for all elementary tensors a® b EA 8 B). One can deduce this from
Takesaki's Theorem (since Proposition 3.2.3 implies that ® is a cross norm
and ®max is subcross). However, we give an independent proof.
Lemma 3.4.10. If II· Ila is a C*-norm on A 8 B, then it is a cross norm.Proof. Let A®aB C JB(H). By the C-equation, it suffices to show JJa@bJJ =
1 for positive norm-one elements a E A and b E B. Let c > 0 be given and let
e = X[l-c:,l] (a) and f = X[i-c:,l] (b) be spectral projections. The projections
e and f commute (since they belong to the weak closures of the ranges of
the commuting restriction maps) and we claim that ef =/= 0. To see this,
note that functional calculus provides nonzero elements ao E C (a) and
bo E C*(b) such that ao:::; e and bo :::; f; hence 0 =/= ao ® bo:::; ef.