1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

154 4. Constructions

Exercise 4.6.5. Let Ji be a C*-correspondence over A and assume that the

quotient map from A onto A/ I1i is locally liftable. Prove that the quotient

map from T(H) onto O(Ji) is locally liftable. (Hint: Use Theorem C.4.)

4.7. Reduced amalgamated free products

Thanks to Voiculescu's groundbreaking noncommutative probability theory,

the free product construction is all over the C* -literature these days. Our

only goal (realized in the next section) is a theorem of Dykema which asserts

that reduced free products preserve exactness. As with Pimsner's construc-

tion, a lot of technical preliminaries are required, and for the remainder of

this chapter most sentences should be understood as nontrivial exercises.

Construction and freeness. Let Ai, i EI, be unital C-algebras, each of
which contains a unital copy of some fixed C-algebra D. Assume that for
each i there exist nondegenerate conditional expectations Ei from Ai onto
D. In this situation, one can construct the reduced amalgamated free product
C-algebra n(Ai, Ei)· The simplest, though highly nontrivial, case is when
D = <C and Ei are states.
With 1 E D c Ai and Ei: Ai -+ D as above, we denote by ( 1ri, Jii, ei)
the GNS representations for (A, Ei)· Note that each 'Tri is faithful, because
of nondegeneracy. The Hilbert D-submodule eiD c Hi is isomorphic to D
and Jii ~ eiD EB Jif. We recall that Af = Ai n ker Ei satisfies D Af D c Af
and Jif = Afei is a C-correspondence over D. We define the free product
Hilbert D-module (Ji, e) = (Jii, ei) by

Ji= eD EB EB EB Jif1 ®n ... ®n Jifn.

n~l ii#···#in

Here, eD is the trivial Hilbert D-module D with e = :i and the notation
"i1 i= · · · i= in" means that i1 i= iz, iz i= i3, ... , in-I i= in. For each i E I,
let

EB
n~l i#i1
ii #···#in
and define an isomorphism Ui E JIB(Jii ®n Ji(i), Ji) by the identifications

eiD ®n eD eD

ui: Jif ®n eD ----+ ee Jif

eiD ®n (Jif 1 ®n · · · ®n 1ifJ 1if 1 ®n · · · ®n Jifn

Ji? i ®n (Ji? ii ®n · · · ®n 1i^0 in ) Ji? i ®n Ji? ii ®n · · · ®n Ji? in•

We define a -representation Ai: Ai-+ JIB(Ji) by Ai(x) = Ui(1ri ® l)(x)Ut.
Observe that Ai[D is the canonical left action of Don the C-correspondence
Ji over D, and hence Ai/D = Aj[D for every i and j.