1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.3. Nuclear and exact C*-algebras 33

Diagrammatically, a C -algebra is exact if there exist a faithful repre-
sentation 7r: A__,. IIB(H) and diagrams
A ___ 1f -~ IIB(H)
' ' /if
' /
'Pn '-°" / / 'l/Jn
Mk(n)(CC)
which asymptotically commute pointwise. Exact C-algebras are sometimes
called nuclearly embeddable. Note that a C -algebra is exact if and only
if there exists a nuclear embedding into some C -algebra (since we could
faithfully represent that C-algebra to get a faithful nuclear representation).
Returning to the issue of dependence on the range, let 7r: A__,. JIB(1i) be
a faithful representation; then, A is nuclear if and only if 7r is nuclear when
regarded as taking values in 7r(A), while A is exact if and only if 7r is nuclear
when regarded as taking values in IIB(H). In particular, nuclearity implies
exactness (the converse is false).
It turns out that the von Neumann algebra analogue of exactness is
slightly tricky to formulate. If one gives the obvious adaptation of Definition
2.3.2, then Proposition 2.1.4 would imply that every von Neumann algebra
enjoys this property (and hence this isn't the right W-notion). We will
study the proper analogue in Chapter 14. However, nuclearity is easily
adapted to von Neumann algebras as follows.
Definition 2.3.3. Avon Neumann algebra Mis called semidiscrete if the
identity map idM: M __,.Mis weakly nuclear.

A fundamental theorem of Connes states that for separable factors the
notion of semidiscreteness is equivalent to numerous other conditions includ-
ing injectivity and hyperfiniteness. We will discuss this deep and important
result later; it turns out to have important consequences in the theories of
nuclear and exact C*-algebras (see Theorem 9.3.3). We have a more modest
goal at the moment, namely, the observation that semidiscreteness of the
double dual implies nuclearity of the algebra. The proof requires two simple,
but very useful, lemmas. The first is a general functional analytic fact.
Lemma 2.3.4. Let A be a Banach space, IIB(A) be the space of all bounded
linear maps from A to A and Cc IIB(A) be any convex set. Then the point-
weak and point-norm closures of C coincide.

Proof. Let T E IIB(A) be a map such that there exists a net {TihEI CC

with the property that for every a E A and functional rJ E A*,

rJ(Ti(a)) __,. rJ(T(a)).^4

4In other words, assume T belongs to the point-weak closure of C.