Weakly Exact van
Neumann Algebras
Chapter 14
As noted earlier, if one defines exact van Neumann algebras as those which
have a weakly nuclear embedding into IBl(H) - i.e., the obvious adaptation
of our C*-definition - one has defined nothing new (since every c.c.p. map
into IBl(H) is weakly nuclear). However, there is a sensible alternative, based
on tensor products, which we now briefly explore.14.1. Definition and examplesThroughout this chapter, B will be an arbitrary unital C*-algebra and J <JB
will be an ideal. The canonical quotient map will be denoted by Q : B ---*
B / J. All C* -algebras, except ideals, are assumed to be unital.
Recall that a C* -algebra A is exact if there is a canonical identification
B@A rv
JQ9A =(B/J)0A,for any J <J B (Theorem 3.9.1). One may rephrase this result as follows:
A is exact if for any J <J B and any -representation 7r: A 0 B -t IBl(H)
with A 0 J c ker 7r, the induced representation ?f: A 0 (B / J) --- IBl(H) is
min-continuous. Inserting the adjective normal in properly, we have our
W* -definition.
Definition 14.1.1. A van Neumann algebra Mis said to be weakly exact
if for any J <J B and any left normal *-representation^1 7r: M 0 B -t IBl(H)
(^1) Recall that 7r is left normal if 7rlM@Cl is normal.
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