Local Reflexivity and
Other Tensor Product
Conditions
Chapter 9
This chapter introduces yet another finite-dimensional approximation prop-
erty - local reflexivity - which is inspired by classical Banach space theory.
Reformulating in terms of tensor products leads to three closely related no-
tions: Properties C, C' and C". It turns out that the first two are equivalent
- and equivalent to exactness -and both imply property C", which ends up
being equivalent to local reflexivity. Consequently we arrive at Kirchberg's
deep discovery that exactness implies local reflexivity, the main result of
this chapter. It will follow that exactness passes to quotients, which might
not sound exciting, but it is an important and very difficult permanence
property.
These results are the culmination of years of hard work, by many able
hands, and represent some of the deepest and most difficult theorems in C -
algebra theory (e.g., they depend on some of the deepest and most difficult
theorems in von Neumann algebra theory). We have tried to keep the pre-
ceding chapters reasonably self-contained in the sense that theorems quoted
without proof either could be found in introductory texts or are only used
for results of peripheral interest. Unfortunately, that is not possible in the
present chapter without adding many pages of hardcore W -exposition. We
will need the fact that the double dual of a nuclear C* -algebra is semidiscrete
and presently there is no proof of this which avoids Connes's thesis (and the
Tomita-Takesaki theory that preceded it). Hence we state, without proof,
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