1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

222 6. Amenable Traces

Tu(h.>.. ·)are the corresponding states, then the net ('1.b.>..)IM converges to Tin

the weak topology on M*. Hence, by the Hahn-Banach Theorem, we may

assume that 11(1/!.>..)IM-TllM. ----r 0 and llh.>..-Uh.>..u*IJ1,Tr ----r 0 for u E U(A). It

follows that the net <pn in condition (2) can be taken so that llT-tr 01.pnllM. ----r

0. (In fact, one can arrange T = tr o<pn with sufficient effort.) For a countable
   ultraweakly dense subset S in the unit ball of A, we can find a subsequence
   'Pn(k) of 'Pn such that ll'Pn(k)(ab) - 'Pn(k)(a)'Pn(k)(b)IJ2,tr ----r 0, for all a,b ES
   and a fortiori for all a, b E A.
   Exercises

Exercise 6.2.1. Observe that every trace on an injective von Neumann

algebra is amenable.

Exercise 6.2.2. Let B = Mn(I)(C) E9 Mn( 2 )(C) and define a trace on B by

p q-p

T(T E9 S) = -tr(S) + -tr(T),

q q

where p < q EN. Show that there exists a unital embedding BC Mn( 3 )(<C)

such that tr (on Mn( 3 )(<C)) restricts to T (on B). In other words, there is a

trace-preserving embedding (B, T) C (Mn( 3 )(CC), tr).

Exercise 6.2.3. Prove that if B is any finite-dimensional C* -algebra and T

is a trace on B, then for every E > 0 there exists a matrix algebra Mn(C)

and a unit al *-homomorphism 7r : B ----r Mn ( <C) such that

for all b E B. Hence, there is nothing gained by replacing matrix algebras
in part (2) of Theorem 6.2.7 with general finite-dimensional C*-algebras.^9

Kirchberg first used the terminology "liftable" in reference to what we've
called amenable traces. The following exercise explains his point of view.
(See Appendix A for the ultraproduct Rw of the hyperfinite II 1 -factor R.)

Exercise 6.2.4. Show that a trace Tis amenable if and only if there exists
an embedding

7r7(A)" c Rw

such that the *-homomorphism 7r 7 : A ----r 7r 7 (A)" c Rw has a u.c.p. lifting
A ----r f^00 (R). There are a couple of different proofs that one could give.
Perhaps the most elegant is to use injectivity of f^00 (R) and the last condition
in Theorem 6.2.7.

(^9) 0n the other hand, it can be convenient to use finite-dimensional algebras in proofs and
this exercise implies that this is legal.