1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

6.3. Some motivation and examples 223

In [41] another important characterization of amenable traces was given;

it also illustrates the connection with classical amenability. In the literature

this is sometimes called a F¢lner condition and has been very useful. For

example, Sorin Popa used it to significantly simplify the proof of "injective

implies hyperfinite" for von Neumann algebras with a faithful tracial state.

Exercise 6.2.5. Let A C JE('H) be a representation which contains no

nonzero compact operators and let T be a tracial state on A. Show that

T is amenable if and only if there exist finite-rank projections Pn E JE('H)

such that

and
Tr(aPn)
Tr(Pn) -+ T(a),
for all a E A. (Hint: For the "only if'' part we can use Voiculescu's Theorem
to pull the u.c.p. maps appearing in statement (2) of Theorem 6.2.7 back
to the given representation A c JE('H), since we assumed A contains no
compacts.)

6.3. Some motivation and examples

The first point which should be clarified is why amenable traces are ana-
logues of invariant means on groups. Recall that a state cp on goo (r) is called
an invariant mean if it is left translation invariant - i.e., cp(s.f) = cp(f) for
all s E r and f E goo (r). In other words, invariant means are just states on
a von Neumann algebra which are invariant under a particular group action.
Fix a concrete representation A C JE('H) of a unital C -algebra and let
U(A) denote the unitary group of A (with the discrete topology). There is
a natural action of U(A) on JE('H) given by conjugation: T 1-+ uTu, for all
TE JE('H) and u E U(A). Hence the following formulation of amenable trace
is equivalent to Definition 6.2.1.

Definition 6.3.1. Given Ac JE('H), an amenable trace on A is the restric-
tion (to A) of a state on JE('H) which is invariant under the conjugation
action of U(A) on JE('H).

Unlike the case of groups, there is really no canonical faithful embedding
A c JE('H) (unless one goes to the nonseparable universal representation).
However, Proposition 6.2.2 states that being an amenable trace is indepen-
dent of the choice of faithful representation and hence the lack of a canonical
action of U(A) on JE('H) doesn't cause problems with the notion of amenable
trace.