1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

162 4. Constructions

Since A is a quotient of a C* -subalgebra of pC* (Ai EBA2, T)p, the composition

of these maps is the desired inverse of p. D

Finally, we deduce the existence of free products of c.p. maps.

Theorem 4.8.5. Let 1 E D C Ai and nondegenerate conditional expec-

tations Ef from Ai onto D be given. Assume 1 E D c Bi and assume

there exist nondegenerate conditional expectations Ef from Bi onto D. Let

Bi: Ai---+ Bi be u.c.p. maps such that (Bi) ID= idD and Ef oei = Ef. Then,

there is a u.c.p. map

8: (Ai,Ef) *D (A2,E2'1)---+ (Bi,Ef) *D (B2,Ef)

such that 8ID = idD and

8(ai .. ·an)= ei1(ai) ... ein(an)

for aj E Aij with ii =/= · · · =/= in.

Proof. By Theorem 4.8.2, we have unital complete order embeddings
(Ai, Ef) D (A2, E2'1) C PAC(A1 EB A2, TA)PA,
(Bi, Ef) D (B2, Ef) C PBC(Bi EB B2, TB)PB·
We claim that the restriction of 8: C(AiEBA2, TA) ---+ C(BiEBB2, TB) given
by applying Proposition 4.6.23 to Bi EB 02 is the desired u.c.p. map. Since
PA and UA are polynomials in TA and TA, for aj E Aij with ii =/= · · · =/= in,
we have
WA(ai ···an)= f(TA,TA,a1, ... ,an),
where f(TA, TA_, ai, ... , an) is a linear combination of monomials in which
each ai, ... , an appears once in this order. Since TA_aTA = a(EJ5(a)) = 0
for all a E Ai. J and AiA2 = {O}, every monomial is of the form

Tmo aiTm1 ... Tmk-1 ak (T)mk ... an (T)mn.

The same holds for W B ( Bi 1 ( ai) · · ·Bin (an)). Hence, we have

8(WA(a1 ···an))= f(TB, T!B, Bi 1 (a1), ... , Bin(an)) = WB(Bi 1 (a1) · · · Bin(an))

by Proposition 4.6.23. D

We have seen that nuclearity typically is not preserved by free products
(e.g., C~(JF2) = C('JI') * C('JI')). However, when dealing with pure states,
things are different. We need a technical lemma of Kishimoto and Sakai.

Lemma 4.8.6. Let A be a unital nuclear C* -algebra, ( 7f, H) be an irreducible
representation, K C 1t be a finite-dimensional subspace and PK E lll(H) be
the orthogonal projection onto K. Then, there exist nets of u.c.p. maps
an: A---+ Mk(n)(C), f3n: Mk(n)(C)---+ A and isometries Vn: K---+ .e~(n) such
that the net f3n o an converges to the identity of A in the point-norm topology