1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

218 6. Amenable Traces

and, by the Powers-St0rmer inequality,

1 1 1 1/2 1 2 1

l -Tr(h2uh2u*) = 2JJh -uh2u*Jb :S 2JJh-uhu*J[1.

We use here and there the fact that Tr(ST) = Tr(S^112 TS^112 ) 2: 0 for any

positive operators S, T with S finite-rank. Now, let's compute tr(cp( u)cp(u*)):

tr ( cp ( u ) cp ( u )) = Tr(P(u@ Tr(P) l)P(u@ l)P)

l:~j=l Tr(QiuQju*) Tr(PiPj)

q

= t min{pi,Pj} Tr(QiuQju*)

i,J= ..^1 q

k

2: Tr(h~uh~u*) - L [Pi; Pj[ Tr(QiuQju*)

i,J=.. 1 q

since min{pi,Pj} = ~(Pi+ Pj - IPi - Pjl) 2: y'PiPj-~[Pi - Pjf· We have

further that

k k k k

L Pi Tr(QiuQju*) = L Pi Tr(Qiu(L Qj)u*)::::; L Pi Tr(Qi) = 1

i,j=l q i=l q j=l i=l q

and hence

k

L [Pi; Pjl Tr(QiuQju*)

i,j=l q

~ ,tl ( 1Pf 2:;J I Tr( Q,uQ;u') l) (IP! ;JI Tr( Q,uQ ;u') l)

1 1

::::; (t IPf ;P}I' Tr(Q,uQ;u')) 2 (t Jpf -pJJ2 Tr(QiuQju*)) 2

i,J= ..^1 q i,J= ..^1 q

s: (2+2Tr(Zluhlu•J )' ( 2 -2Tr(hluhlu•))l

:S [[h-uhu*JJi^12 ,

by the previous computations. Finally, combining everything, we get

k
tr(cp(uu) - cp(u)cp(u)) ::::; 1 -Tr(h^112 uh^112 u) + L IPi; Pj[ Tr(QiuQju)
i,j=l q