1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.8. Maps on reduced amalgamated free products 161

But since Ai 1 ( ai) · · · Ain ( an)~D C 1-lf, 1 @v · · · @v Hin and in -/= j, we have

that

P'H(j) Ai 1 ( ai) · · · Ain (an) P~v = 0.

Therefore, in both cases, we have

<Pi(Ai 1 (ai) · · · Ain+l (an+i)) = <Pi(Ai 1 (ai) · · · Ain(an))<Pi(Ain+l (an+i)).

Thus induction will complete the proof of the claim.

Now ( <>) follows from this claim and the preceding calculation. The same

holds for Ki,4·

Finally, we set ']! = EE\z wi,l: A ---+ JB(JC,). Then, w is a u.c.p. map

into pC*(Ai EB A2, T)p with the desired properties. We can define a *-

homomorphism 7f: C*(w(A))---+ A by 7r(x) = Vi,ix'Vi~i.E JB(H). D

Here are three important consequences. The first states that exactness
is preserved by reduced free products.

Corollary 4.8.3. Let 1 E D C Ai ( i E J) be unital C* -algebras with non-

degenerate conditional expectations Ei from Ai onto D. Then, the reduced

amalgamatedfree product (A,E) = *v(Ai,Ei) is exact if every Ai is exact.

Proof. Since the amalgamated free product construction is associative and
exactness is closed under inductive limits, we may assume that I= {1, 2}.
By Theorem 4.6.25, the C-algebra C(Ai EB A2, T) ~ T(H~iEBA 2 ) is exact
if Ai EB A 2 is. Since A= Ai D A2 is a quotient of C(w(A)) c pC(Ai EB
A 2 , T)p with the u.c.p. splitting w, exactness of A follows from that of
C(Ai EB A2, T). D

The following corollary is rather easy if the conditional expectations are
faithful.

Corollary 4.8.4. Let 1 E D c Ai and conditional expectations Ei from Ai

onto D be given. Let 1 E DB C Bi C Ai be such that Ei(Bi) =DB. Assume

Ei and Ef = EilBi are nondegenerate. Then,

(Bi, Ef) *Ds (B2, Ef) ~ C*(Bi, B2) c (Ai, Ei) *D (A2, E2).

Proof. Let A = (Ai, Ei) D (A2, E2) and B = (Bi, Ef) Ds (B2, Elf).
Because of freeness, it is clear that the GNS representation p of C(Bi, B 2 )
with respect to the conditional expectation E of A yields B. We have to
show that p is faithful. If E is faithful, then this is trivial;· otherwise, it
is not so simple. This is why we need Theorem 4.8.2. We construct the
inverse of p as follows. The C -algebra B is completely order isomorphic to
the C-subalgebra of pC(Bi EBB2, T)p. But pC(Bi EBB2, T)p is canonically
-isomorphic to the C-subalgebra of pC (A1 EB A2, T)p by Corollary 4.6.21.