1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

52 2. Nuclear and Exact C* -Algebras

Hence we have

E f

1

IF(μ, r)jdr = E > L llsμ - μ111 = f

1

L lsF(μ, r) L F(μ, r)jdr.

Jo sEE Jo sEE

Thus for some r we must have

L lsF(μ, r) L F(μ, r)I < clF(μ, r)I,

sEE

which shows that F(μ, r) is almost invariant under translation by the ele-

ments in E.

(3) * (4): Let (Fi) be a F¢lner sequence and ei = IFi1-^1 l^2 xFi be the

normalized characteristic functions of the Fi's (viewed as unit vectors in

£^2 (r)). The same calculation used in the £^1 context (see the paragraph after

Definition 2.6.4) shows that ii>-s(ei) - eille2(r) -+ 0 for every s EI'.

(4) * (5): Consider the vector states x ~ (xei,ei)· As noted in the

previous section, these restrict to positive definite functions on I' and ob-

viously tend to 1 pointwise. To make them finitely supported, one simply

forces each ei to be a finitely supported £^2 function.

(5) ==?-(6): Take a net ('Pi) as in condition (5). By Theorem 2.5.11, the

multipliers m'Pi (resp. mcpi) are u.c.p. on C*(I') (resp. Ct(r)). We note that

>. o m'Pi = m'Pi o >. on C* (I') since the two maps are continuous and coincide

on the dense subspace qr]. Observe that mcpJx) -+ x for every x E C*(I')

since this is true for x E qr]. Now suppose x E C*(I') and >.(x) = 0. Then,

we have

>.(m'Pi(x)) = mcpJ>.(x)) = 0

for every i. But since 'Pi is finitely supported, we have m'Pi ( x) E qr], and

hence >.(mcpJx)) = 0 implies mcpJx) = 0. Therefore, x = limimcpi(x) = 0

and the *-homomorphism>.: C*(I')-+ Ct(r) is injective.

(6) ==?-(7): The trivial representation I'-+ iC extends to C*(I') = Ct(r).

(7) ==?- (1): Let r: Ct(r) -+ iC be any *-homomorphism, but regard it

as a state. Extending to JB(C^2 (I')), we may assume that r is also defined on

C^00 (I') c JB(C^2 (I')). Since the left translation action is spatially implemented,

r(s.f) = r(Asf >.;) = r(>.s)r(f)r(>.s) = r(f)

for all s E I' and f E £^00 (I') (the unitaries As belong to the multiplicative

domain of r). Hence, r is an invariant mean as desired.

At this point we have shown the first seven conditions to be equivalent.

(4) {:} (8): The==?-direction is easy. For the converse, it suffices to show
that if E is a finite symmetric set (meaning E = E-^1 ) satisfying condition
(8), then E generates an amenable group. In this situation, the norm-one