4.8. Maps on reduced amalgamated free products 157Exercises
Exercise 4.7.1. Let (A, E) = D(Ai, Ei), where 1 ED C Ai are unital C-
algebras with faithful conditional expectations Ei from A onto D. Prove
that E is faithful on A. (Hint: For the natural isometry Vi E IB(1-li, 1-l), we
have Vi AVi =Ai. Assume that x EA is given such that x(ai®· ··®an) of. 0,
and prove that x(ai ® · · · ®an-i) of. 0.)
Exercise 4.7.2. Let (A,E) = D(A,Ei), where 1 ED c A are unital
C* -algebras with nondegenerate conditional expectations Ei from Ai onto
D. Suppose that there exists a tracial state TD on D such that every TD o Ei
is tracial on Ai· Prove that TD o Eis tracial on A. (Hint: Induction on the
length of words.)
4.8. Maps on reduced amalgamated free products
Let 1 E D c A, i = 1, 2, be unital C*-algebras with nondegenerate condi-
tional expectations Ei from A onto D. Consider the u.c.p. map
cp: Ai EB A2 3 (ai EB a2) 1--t E2(a2) EB Ei{ai) ED EB D c Ai EB A2
on Ai EB A2 and the C* -correspondence 1t~iEElA 2 over Ai EB A2 that is given
in Example 4.6.11. We will describe a faithful representation of T(1t~iEElA 2 ).
Let Hi = L^2 (A, Ei) and ~i = :CJ: E 1-li· Consl.der the Hilbert D-module
JC, =Ki EB K2, where
EB
Let T E IB(JC) be the isometry which exchanges lCi and lC2, and acts like a
creation operator:
T( (i ® · · · ® (n) = ~i ® (i ® · · · ® (n,
where i = 1 or 2 is suitably chosen. We have T* (i = 0 and
T*((i ® · · · ® (n) = (~iu (i/(2 ® · · · ® (nfor n 2:: 2. We regard Ai as a C -subalgebra of lffi(JC,i) acting on the first
tensor component. Thus, Ai EB A2 C IB(JC). It is easily seen that
T(ai EB a2)T = E2(a2) EB Ei(ai)
for every ai EB a 2 E Ai EB A2. Since the family of unitary operators Uz =
EBn>i zn implements the gauge action and the compression to Hi EB1t2 C JC
separates Ai EB A 2 from the "compact operators" span((Ai EB A2)T(Ai EB
A 2 )T(Ai EB A 2 )), the gauge-invariant uniqueness theorem implies:
Lemma 4.8.1. The C-algebra T(1t~iEElA 2 ) is canonically -isomorphic to
the C -subalgebra C* (Ai EB A2, T) of IB(JC).