156 4. Constructionsalgebraic amalgamated free product^12 of Ai's over D and let 7r: A ----+ A
and 7r^1 : A----+ A' be the canonical representations. Because of conditions (1)
and (2) and uniqueness of the GNS representation, it suffices to show that
E' o 1f^1 =Eo7r. Since Ai= D + Af and DAf DC Af, we haveA= span(D U LJ LJ Af 1 · · ·AfJ.
n2'.:1 i1 =f···=fin
Hence, property (3) implies that x - (E o 7r)(x) E span(LJ Af 1 • · · AfJ for
every x E A. It follows that (E' o 7r^1 )(x) = (E' o?r')( (E o 7r) (x)) = (E o 7r) (x ),
again by condition (3), and we are done. 0
Example 4. 7.4. An inclusion of discrete groups A :SI' gives rise to an in-
clusion C~(A) c C~(r), and there is a conditional expectation EI: C~(r)----+
C~(A) (Corollary 2.5.12).
Let I' = AI'i be an amalgamated free product of discrete groups I'i
over a common subgroup A (cf. Appendix E). Then, we have
(C~(r), EI)= q(A)(C~(ri), Eii).
This is easily checked by verifying the conditions of Theorem 4. 7.2 and in-
voking uniqueness. It is also useful to see the isomorphism at the Hilbert
space/C-module level. As an important special case, we have natural iso-
morphisms C~(lFn) ~ C(Z) · · · C(Z) - the n-fold free product, taken
with respect to the canonical trace on C(Z) - for all n EN U { oo }.
Example 4.7.5 (Speicher). Let Hi be C*-correspondences over A. Then,
we have
Proof. We will show the T(Hi)'s are free over A in (T(ffi Hi), EEBr-tJ
(which suffices, by Theorem 4.7.2). Since
ker E'l-li = span{TμT; : μ E H?m, v E H?n with (m, n) -=J (0, O)},this reduces to showing
EEB 1-li (Tμ1 r:l ... Tμk r:k) = 0
for μj E 1-{~mj J and Vj E 1-{~nj J with (mj, nj) -=/=: (0, 0) and ii -=/=: · · · -=/=: ik.
But this easily follows from the facts that EEB 1-li (TtJ = 0 and TtT'T/ = 0,
whenever ~ E Hi and rJ E 1-lj with i -=J j. 0
(^12) That is, the universal algebra generated by copies of Ai which agree on D. It has the
universal property that if 13 is any algebra containing a unital copy of D and Ai -> 13 are D-
preserving homomorphisms, then they extend uniquely to a homomorphism from A into 13. If you
prefer, here's a categorical definition: An amalgamated free product over D is the coproduct in
the category whose objects are algebras containing unital copies of D and whose morphisms are
D-preserving homomorphisms.