1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

32 2. Nuclear and Exact C*-Algebras

Since the sum of positive operators is positive, it is easy to check that the

7/Jn's are u.c.p. Since bn -+ 0, it follows that 7/Jn o 'Pn -+ (} in the point-

ultraweak topology. D

Proposition 2.2.8. Let M and N be van Neumann algebras and (}: M -+

N be a unital weakly nuclear map. Then, there exist normal u.c.p. maps

'Pn : M -+ Mk( n) (CC) and 7/Jn : Mk( n) (CC) -+ N such that 7/Jn o 'Pn -+ (} in the

point-ultraweak topology.

Proof. Fix a finite set~ C M, a finite set of normal functionals x C N*

and E > 0. By the last proposition, we can find u.c.p. maps <p: M-+ Mk(<C)

and 'ljJ: Mk(CC) -+ N such that

l11(e(m)) - ry('ljJ o c,O(m))I < c

for all m E ~ and 17 E x. Of course 'ljJ is automatically normal so we only

have to replace cp with a normal u.c.p. map.

By Corollary 1.6.3 we can find a net of normal u.c.p. maps 'P>..: M-+

Mk(<C) which converge to <pin the point-norm topology. This completes the

~~ D

2.3. Nuclear and exact C*-algebras

Definition 2.3.1. A C*-algebraA is nuclear if the identity map idA: A-+ A

is nuclear.

Sometimes nuclear C* -algebras are called amenable, which comes from a

cohomological characterization ([42], [76]); sometimes they are said to have

the completely positive approximation property (CPAP), which refers to the

existence of diagrams

A !<lA A

' ' / if

'Pn'-~ ' //'I/Jn /

Mk(n)(CC)

which asymptotically commute pointwise. Though never convenient when
writing, it is often very helpful to draw diagrams as above. For example,
Arveson's Extension Theorem gets used all the time and sketching diagrams
helps one understand which maps can be extended, where the extensions
are defined and where they take their values. (For example, Exercise 2.1.9
is completely transparent if one draws the right diagram.)

Definition 2.3.2. A C*-algebra A is exact if there exists a faithful repre-
sentation 7r: A-+ JB(H) such that 7r is nuclear.