1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.8. Maps on reduced amalgamated free products 159

where the @ is just a marker to indicate when the first eiD in the middle

appears, and the direct sum is taken over the positions of®· Finally,

Ki,4 =Hf ®n ejD ®n Ki EB E!jHi ®n Hj ®n · · · ®n Hf ®n ejD ®n Ki
consists of those tensor products such that ejD appears before eiD in the
middle.
We observe that the (partial) isometry u interchanges JC,i,i (resp. JC,i,3)
and Kj,2 (resp. Kj,4) for i i= j and that each JC,i,l is invariant under 'lj;k (a),
for any i, k and l.

Via standard identifications eiD ®n Hj ~ Hj and Hj ®n eiD ~ Hj, we
have a natural isomorphism Vi,i: Ki,i , H, where (H, e) = (Hi, 6)(H2, 6)
is the free product Hilbert D-module. We let A= Ai D A2 and define a
u.c.p. map wi,i: A, IIB(Ki,i) by wi,i(x) = Vi:ixVi,i. It is routine to check
that

Vi:iAi(ai)Vi,i = (ai)JKi,1 = 'lf;i(ai)!Ki,1'

Vi:iAj(aJ·)Vi,i = (uaju)JKi,1 = 'lf;j(aj)IKi,1

for every ai E Ai and aj E Aj (i i= j). Therefore, Wi,i is a *-isomorphism
such that

Wi,i ( ai · · ·an) = 'lf;i 1 ( ai) · · · 'lf;in (an) JKi,l

for any ak E Aik· The same holds for Ki,2·

Let
H(j) = E9 E9 Hf 1 ®n · · · ®n Hfn CH.
n;:::i il# .. ·#in
in=j
Via the identification eiD ® D Hj ~ Hj' we have a natural isomorphism

Vi,3 : JC,i,3 ____, H(j) ® D eiD ® D J(,j c H ® D ( eiD ® J(,j).

Define wi,3: A____, IIB(Ki,3) by Wi,3(x) = Vi: 3 (x @l)Vi,3 and we will prove that

(<>) wi,3(.Ai 1 (ai) · · · Ain(an)) = 'lf;i1 (ai) · · · 'lf;in(an)JKi,s

for ak E Afk with ii i= · · · i= in. Let a E Af and b E Aj be given (i i= j).
With the help of Lemma 4.7.3, we first check, for (k E Hfk with i =ii i=
i2 i= · · ·, that

Wi,3(.Ai(a))((i ® (2 ® · · ·@ ei ® · · ·)

= Vi:3 ( ( (a(i - ei(ei, a(i)) ® (2 + (ei, a(i)(2) ® ... ® ei ® ... )

= a(i ® (2 ® ... ® ei ® ...

= p(a(i ® (2 ® ... ® ei ® ... )

= 'lj;i (a) ( (i ® (2 ® ... ® ei ® ... )