- Reduced amalgamated free products 155
Definition 4.7.1 (Reduced amalgamated free product). The reduced amal-
gamated free product (A,E) = *D(Ai,Ei) is the C*-subalgebra A of JB('H)
generated by LJiEJ Ai(Ai), together with the conditional expectation from A
onto D given by E(x) = (e, xe).The construction * D is commutative and associative, if you can fathom
the notation required to check this. Direct calculation shows that Ei = Eo>.i
on A. Also, Ai is injective (because Ei is nondegenerate); hence we often
omit ~ and view Ai as a subalgebra of A. Here are the main properties of
the reduced amalgamated free product:
Theorem 4.7.2. Let 1 ED C Ai be unital C*-algebras with nondegenerate
conditional expectations Ei from Ai onto D, and let (A, E) = * D (Ai, Ei).
(1) There is an inclusion 1 E D C A and a nondegenerate conditional
expectation E : A -+ D.
(2) There are inclusions D c Ai CA which are compatible on D, and
A is generated by LJ Ai as a C* -algebra.
(3) One has EIAi = Ei for every i, and the C* -subalgebras Ai are free
over D in (A, E). Namely,
E(ai ···an)= 0
for every aj E Af. J with ii "!-· · · "!-in.
Moreover, the above conditions uniquely characterize the reduced amalga-
mated free product (A, E).To verify the third assertion, we need the following fact; sanity prevents
us from TeXing the proof.
Lemma 4.7.3. Let a E Ai and (j E 'Hi. J with ii"!-···"!-in. Then,
{(a(i - ei\ei, a(i)) Q9 (2 Q9 .•. Q9 (n
a((i Q9 · · · Q9 (n) = +(ei, a(i)(2 Q9 · · · Q9 (n
aei Q9 (i Q9 · · · Q9 Cnif ii= i,
(When ii = i and n - 1, the term (~i, a(i)(2 should be understood as
~(~i,a(i).)
In passing, we note that there is a conditional expectation from A onto
Ai which preserves E. Indeed, if Vi E JB('Hi, 'H) is the natural isometry given
by Vi~i = ~ and Vi( = ( for ( E 'Hi, then x r--t Vi*xVi defines the desired
conditional expectation from A onto Ai.
Proof of Theorem 4.7.2. We only prove the last assertion. Let (A',E')
be another pair which satisfies conditions (1)-(3). We denote by A the