398 14. Weakly Exact von Neumann AlgebrasBy assumption, 7r extends to a bi-normal *-homomorphism fr on M ® B**.
Since M ®Jc ker7r, we have fr(l ® p) = 1. It follows that 7r coincides with
fro (id® 'ljJ) and hence is min-continuous. Indeed, for any a E M and x E B,
we have
fro (id® 'lfJ)(a ® Q(x)) = fr(a ® px) = 7r(a ® x) = 1f(a ® Q(x)).We next prove the "only if" direction; it's almost the same as Proposi-
tion 9.2.7. For a given B, let I, B1 and O": B1 ~ B** be as in the proof of
Proposition 9.2.7. For a given left normal representation 7r: M@B ~ JE(1i),
we consider the representation p: M ® B1 ~ JE(1i) defined by
n n
p(:[ ak ® (xk( i) )iEJ) = strong*-lii~n 7r(l: ak ® Xk( i) ).
k=l i k=l
Since M is weakly exact, p is left normal and M ® ker O" c ker p, the induced
representation p of M 0 B** is min-continuous. Since fr = p, this completes
the proof. DThe advantage of the above formulation is that it can be generalized to
the case that Bis not a C-algebra. This fact is important, so let's be more
precise. Let X be a C -algebra or an operator space. Let p E M be the
central support of the identity representation of M so that we may identify
M with pM. Then, the canonical bi-normal embedding M 0 X C
(M ® X) gives rise to a (nonunital) bi-normal embedding
8 x : M 0 X = pM 0 X <-+ ( M ® X) .
Let a E M and x E X be given and take nets Pi E M and Xj E X such that
Pi ~ p and Xj ~ x in the weak -topology. Then, we have Pia® Xj E M ® X
and
8x(a®x) =weak-lt~(pia®xj) E (M®X).
i,J
Now, assume that Mis weakly exact and let B be a C-algebra. Since 8B
is a bi-normal -homomorphism which is continuous on M ® B, it follows
from Proposition 14.2.1 that 8B is continuous (and isometric) on M ® B.
This implies that 8 x is isometric on M ® X** for any operator space X.
Indeed, when X C B, we have the commuting diagram
M ® X** ----+ 8x (M ® X)**r e r
M®B ~ (M®B),
where the bottom and the vertical inclusions are all isometric. In summary,
we have the following result.