1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

6.2. Amenable traces 219

1 1/2

:::::; 2llh - uhu*ll1 + llh-uhu*ll1 ,

so the proof is complete. D

If w is a state on B, then we define a seminorm II· 112,w on B by llbll2,w =

Jw(b*b).

Lemma 6.2.6. If cp: A----+ B is a u.c.p. map and w is a state on B, then

llcp(ab) - cp(a)cp(b)ll2,w:::::; llallw(cp(b*b) - cp(b*)cp(b))

1

/^2 ,

for all a,b EA.

Proof. Applying GNS, we may assume B c IIB(H) and w = ( · e, e/ for some
e E H. Let 7r: A ----+ IIB(JC) be the Stinespring dilation of cp, with isometry
V: H ----+ JC. Then, for all a, b E A, we have

llcp(ab) - cp(a)cp(b)ll2,w = ll(cp(ab) - cp(a)cp(b))ell

= IJV*7r(a)(1-vv*)7r(b)ve11

:::::; IJV*7r(a)(l - VV*)^1 l^2 1111(1-VV*)^1 l^2 7r(b)ve11

:::::; llall (V*7r(b*)(l - VV*)7r(b)Ve, e/^112

= llallw(cp(b*b) - cp(b*)cp(b))^112.

D

We are almost ready for the main theorem, just a little more notation.
For a tracial stater on A we consider the product *-homomorphism arising
from the left and right regular representations:

Kr x 1f~P: A 8 A^0 P----+ IIB(L^2 (A, r)).

Composing this representation with the vector state x f--+ (xi, i), where i
denotes the natural image of the unit of A, we get a positive linear functional
μ 7 on A 8 A op. Note that for an elementary tensor a ® b E A 8 A op we have

μ 7 (a ® b) = (7r 7 (a)7r~P(b)l, l) = (7r 7 (a)7r 7 (b)l, i) = r(ab),

since 1f~P(b)i = b = 7r 7 (b)i. In the context of residually finite groups, we
already used this functional (see the proof of Proposition 3.7.11); it will
again play a crucial role.

Theorem 6. 2. 7. Let A be unital with tracial state r. The following are
equivalent:
(1) r is amenable;
(2) there exists a net of u. c.p. maps 'Pn: A ----+ Mk(n) (C) such that
r(a) = limtrocpn(a) and ll'Pn(ab) - 'Pn(a)<pn(b)ll2,tr ----+ 0, for all
a,b EA;