286 9. Local ReflexivityIIB('H) and N = ClM ® N ~ IIB('H) are normal representations. Of course, we
may replace IIB('H) with any other von Neumann algebra in this definition.
Proposition 9.2.1. For every A and B, there is a canonical injective bi-
normal map
A** GB**'-----* (A® B)**.Proof. Thanks to the existence of restriction maps (Theorem 3.2.6), the
inclusion AGE'-----* (A®B)** arises from commuting copies of A and B inside
(A® B)**. Thus the weak closures of A c (A® B)** and B c (A® B)**
also commute. Hence there is a bi-normal map
A** GB**~ (A® B)**.
To verify injectivity, one first recalls that functionals of the form cpG'l/J, where
cp (resp. 'lj;) is a linear functional on A (resp. B), separate the elements of
A** GB** (Exercise 3.1.5). By (the proof of) Takesaki's Theorem we can
extend each cp G 'ljJ to a linear functional cp ® 'ljJ on A ® B and then further
extend to a normal functional on (A® B)**, also denoted by cp ® 'lj;. Now
one must check that cp G 'ljJ is the same as
A** GB**~ (A® B)** 'P~'l/J C,
which implies injectivity, as desired. DSince A c A**, we also have a natural inclusion A G B** '-----* (A ® B) **.
Though a bit strange at first, the following definitions are actually quite
natural.
Definition 9.2.2. Let A be an arbitrary C* -algebra.
(1) A is said to have property C if
A** GB**'-----* (A® B)**
is min-continuous for every C* -algebra B.
(2) A is said to have property C' if
AG B** '-----*(A® B)**
is min-continuous for every C* -algebra B.
(3) A is said to have property C" if
A** GB'-----* (A®B)**
is min-continuous for every C* -algebra B.To get acquainted with these notions, let's work out two simple perma-
nence properties.
Proposition 9.2.3. All three of these notions pass to subalgebras.