1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

6.2. Amenable traces 217

Lemma 6.2.5. Let h E JB(H) be a positive, finite-rank operator with ra-
tional eigenvalues and such that Tr(h) = 1. Then there exists a u.c.p.
map rp: JB(H) ---+ Mq(C) such that tr( rp(T)) = Tr(hT) for all T E JB(H)
and I tr (rp(uu) - rp(u)rp(u)) I:::; 2liuhu* -hlii^12 for every unitary operator
u E JB(H).

Proof. First we write

h = Pl Q1 EB p^2 Q2 EB · · · EB Pk Qk

q q q

where P; < P: < · · · < P; are the nonzero eigenvalues of hand Q1, ... , Qk
are the corresponding spectral projections. Notice that

k

LPi Tr(Qi) = q

i=l

since Tr( h) = 1.

Let JC be a Hilbert space and P1 :::; P2 :::; · · · :::; Pk be projections such
that rank(Pi) =Pi for all i. Since the Qi's are orthogonal, we can define a
projection P E JB(H Q9 JC) by

P = Q1 Q9 P1 + Q2 Q9 P2 + · · · + Qk Q9 Pk.

Note that
k
Tr(P) = LTr(Qi) Tr(Pi) = q.
i=l
It turns out that the u.c.p. map rp: JB(H) ---+ Mq(C) we are after is simply
compression by P, i.e., rp(T) = P(T Q9 lK)P for all TE lB(H).

We first check the trace-preserving condition. For TE JB(H) we have

as desired.

tr ( (rp T)) = Tr(P(T Tr(P) Q9 l)P)

l:~=l Tr(QiTQi) Tr(Pi)

q

k

= 2:::-Tr(QiT) Pi = Tr(hT),

i=l q

For a unitary operator u E JB(H), we now compare tr(rp(u)rp(u)) with
1 1
Tr(h2uh2u). Note that