1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

164 4. Constructions

described in Proposition 1.5.12, the c.p. map f3n corresponds to a positive

element an E B(KnEB..Cn)@A**. We note that the c.p. map 7rof3n corresponds

to

(id Q9 7r)(an) E (P!Cn Q9 PJCJ (B(Kn EB ..Cn) Q9 B(1i)) (P!Cn Q9 PJCn).

Since id Q9 7r is an irreducible representation of B(Kn EB ..Cn) Q9 A, it follows

from strong transitivity (Corollary 1.4.8) that a;/^2 E B(Kn EB ..Cn) Q9 A**

is approximated, in the point-ultrastrong topology, by a net (cn,i)i of self-

adjoint elements in B(Kn EB ..Cn) Q9 A such that llcn,ill ::::;; lla;/^2 11+1 and

(id Q9 7r)(cn,i) (1 Q9 PJCn) =(id Q9 7r)(a;/^2 ) (1 Q9 P!Cn).

Then, the positive elements an,i = c~,i in B(Kn EB ..Cn) Q9 A converge to an in

the point-ultrastrong topology and satisfy the equation

(id Q9 7r)(an,i) (1 Q9 PJCn) = (id Q9 7r)(an) (1 Q9 PKn).

Now the c.p. maps f3n,i: B(Kn EB ..Cn) --+A corresponding to an,i converge to

f3n in the point-ultraweak topology and satisfy

7r(f3n,i(x))PJC = 7r(f3n(x))PJC = PK,nXPJC

for all x E B(/Cn EB ..Cn)· This completes the proof. D

Theorem 4.8.7. Let (Ai,'Pi) (i = 1,2) be unital C*-algebras with states 'Pi

whose GNS representations are faithful. Let (A, <p) = (A1, <p1) * (A2, <p2) be

the reduced amalgamated free product C* -algebra. Suppose that both Ai are

nuclear and at least one of the 'Pi 's is pure. Then A is nuclear.

Proof. We may assume that <p1 is pure. By Lemma 4.8.6, there exist nets of

u.c.p. maps an: A-+ Mk(n)(CC), f3n: Mk(n)(CC)--+ A and vector states Wn on

Mk(n) (CC) such that f3n O(Y,n converges to idA, Wn o CXn = <p1 and <p1 o f3n = Wn·

By Theorem 4.8.5, we may define the free product u.c.p. maps

tYn = CXn * idA 2 : A--+ (Mk(n) (CC), Wn) * (A2, <p2)

and En= f3n idA 2 • Since (Mk(n)(CC), wn) (A2, <p2) is nuclear (see Exercise
4.8.2) and En o an = (f3n o an) * idA 2 converges point-norm to idA, we are
done. D

Exercises

Exercise 4.8.1. Let 1 E D c A be C*-algebras with a nondegenerate
conditional expectation E-fJ from A onto D. Let <p = E-fJ : A --+ D c A
and consider T(1i~) as in Example 4.6.11. Prove that there is a natural
isomorphism

(T(1i~), E-f; o EH'P) A ~ (A, E-f;) *D (T(CC) Q9 D, w Q9 idn ),