1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

224 6. Amenable Traces

As usual, group C* -algebras provide some insightful examples. Recall

that the left translation action of r on goo (I') is spatially implemented by

the left regular representation (see Section 2.5).

Proposition 6.3.2. Let r be a discrete group. Then l^00 (I') has an invariant

mean (i.e., r is amenable) if and only if C{(I') has an amenable trace.

Proof. ( :::::?-) Assume 1/; is an invariant mean on goo (r). Consider the condi-

tional expectation <I>: JIB( g^2 (r)) --+ goo (I') defined by

<I>(T) = L e 9 , 9 Te 9 , 9 ,

gEI'

where eg,g is the rank-one projection onto 69 E l^2 (r) and the sum is taken

in the strong operator topology. In the matrix view, <I> simply maps T to

its diagonal (which, of course, is an element in g^00 (r)). A straightforward

calculation shows that

<I>(.AsT>.;) = ,\ 3 <l>(T)>.; = s.<I>(T)

for all TE JIB(g^2 (r)) ands Er. Define a state 'Pon JIB(g^2 (r)) by

tp = 1/; 0 <I>

and we find that

tp(.AsT>-;) = 'l/J(s.<I>(T)) = 'lj;(<I>(T)) = tp(T).

It follows that tp(xT) = tp(Tx) for all x E qr} and density of the group

algebra in C{(r) implies that tp(uTu*) = tp(T) for all unitaries u E C{(I').

( <¢=) The converse is even easier. Assume C{ (I') has an amenable trace

and let tp be a state on JIB(l^2 (I')) such that

tp(uTu*) = tp(T)

for all unitaries u E C{ (I') and T E JIB( g^2 (r)). In particular, if f E l^00 (I'),
then
tp(s.f) = tp(Asf >-;) = tp(J)
and thus the restriction of tp to zoo (I') is an invariant mean. D

Here is a more general result which shows that for reduced group C* -
algebras there is an "all or nothing" principle: Either every trace is amenable
or none are.

Proposition 6.3.3. Let I' be a discrete group. Then the following are equiv-
alent:
(1) r is amenable;
(2) C{(I') has an amenable trace;
(3) every trace on C{(I') is amenable.