1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

214 6. Amenable Traces

Exercise 6.1.6. Let tr be the unique tracial state on Mn(C). Show that

the product map

arising from the left and right regular representations, is an isomorphism.

Exercise 6.1. 7. Show that if rp: A----+ Bis u.c.p., then it is still u.c.p. when

regarded as a map of opposite algebras.

Exercise 6.1.8. Show that if A is nuclear (resp. exact), then A^0 P is nuclear

(resp. exact).

6.2. Amenable traces

In 1975 Alain Connes proved a remarkable theorem which nearly completed

the classification of injective factors with separable predual (cf. [41] - the

one case not covered took a decade, and Uffe Haagerup, to resolve [79]).

Connes's proof was a typical display of deep insight combined with re-

markable technical strength and, in our opinion, ranks among the greatest

achievements in the history of operator algebras. In this section we will ex-

plore one of the ingredients that went into the proof. The main conceptual

idea is that certain traces on C* -algebras are analogous to invariant means

on groups. When viewed this way, some ideas from the theory of amenable

groups can be translated into a more general operator algebraic framework.

Unfortunately having the proper insight is only half of the battle; the

proofs are still very technical and delicate. The main result of this section,

Theorem 6.2.7, is essentially due to Connes in the unique trace case and

Kirchberg in general. We have taken the quickest, most direct route (that

we are aware of) to this result, postponing motivation and examples until

the next section. For some it will be more palatable to skip ahead, take in

the conceptual landscape and then come back for the gory details.

Definition 6.2.1. Let A c lB(H) be a concretely represented unital C*-

algebra. A state T on A is called an amenable trace^3 if there exists a state

rp on JB(H) such that (1) 1PIA = T and (2) rp(uTu*) = rp(T) for every unitary

u EA and TE JB(H).^4

Note that an amenable trace really is tracial. Indeed, if T E JB(H) is

arbitrary and u E A is a unitary, then

rp(Tu) = rp(u(Tu)u*) = rp(uT).

(^3) Beware: Lots of different terminology can be found in the literature. Also, one can extend
the notion to nonunital algebras, but we'll have no need for this.
(^4) The state cp is sometimes called a "hypertrace" or an "A-central state."