226 6. Amenable TracesStatement ( 4) in the previous proposition reveals a subtlety which one
should be aware of: the notion of amenable trace depends on the domain.
Indeed, it can easily happen that r is amenable as a trace on A, but is
not amenable on the quotient A/ J (when the extension is not locally split).
Natural examples of this phenomenon will be explored in the next section
(see Theorem 6.4.3 and the paragraph which follows it). On the other hand,
this problem never occurs for exact C* -algebras.
Proposition 6.3.6. If A is exact and r is a trace on A/ J which is amenable
when regarded as a trace on A, then it is also amenable on A/ J.Proof. We can use the same tensor product argument as above. Indeed,
since A^0 P is exact (Exercise 6.1.8), the sequence
0---+ J 0 A^0 P ---+A 0 A^0 P---+ A/ J 0 A^0 P ---+ 0
is exact and this is all we needed. D
We close this section with the observation that amenable traces are
always a nice subset of the tracial space.Proposition 6.3. 7. The set of amenable traces on A is always a weak-*
closed, convex face in the space of tracial states on A.
Proof. The finite-dimensional approximation property which characterizes
amenable traces can be used to show that amenable traces are closed -just
think about it.
The facial property is a simple application of Proposition 3.8.3. Indeed,
if r = Ar1 + (1-A)r2, where 0 <A< 1, then in the GNS representation for
r we can find a subrepresentation which is unitarily equivalent to the GNS
representation with respect to r 1.^10 Thus the last statement in Theorem
6.2.7 shows that if r is amenable, then so is r 1.
Convexity is also very simple. If AC IIB(H), r 1 and r 2 are amenable traces
with corresponding states <p1 and, respectively, cp 2 on IIB(H) (as in Definition
6.2.1) and 0 < A < 1, then A<p1 + (1 - A)<p2 is a state on IIB(H) (which
evidently extends Ar1 + (1 - .:)r2) and is easily seen to be A-central. D
Exercises
Exercise 6.3.1. Observe that every trace on a C*-algebra with Lance's
WEP (Definition 3.6. 7) is amenable.
Exercise 6.3.2. Observe that the trace on C*(r) coming from the trivial
representation is always amenable.
(^10) If Yr 1 E 7rr(A)' is such that r1(a) = (7r 7 (a)y 71 i, i), then the invariant subspace generated
by Yr 1 i will do.