1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.3. Amenable actions 125

(3) lls *a Ti -7iJl2---* 0 for alls Er, wheres E Cc(r, A) is the function

which sends s 1---+ lA and all other group elements to zero.^4

The functions Ti will replace the F¢lner sets we used in the previous
section.

Lemma 4.3.2. Let A be a I'-C -algebra and T: r ----" A be a finitely sup-
ported function such that 0::::; T(g) E Z(A) for all g Er and Lg T(g)^2 = lA.
Then,
(1) TaT(s) = L T(p)a 8 (T(s-^1 p)), where Fis the support ofT,
pEFnsF
and
(2) IJlA -T a T(s)JI :S llT-s *a Tll2, for alls Er.

Proof. Statement (1) is a trivial calculation, using the fact that T(g)* =
T(g) for all g Er..

To prove the second statement, we first note that S*aT(p) = a 8 (T(s-^1 p))
for all p Er. Now we compute

lA -T *a T*(s) = LT(p)^2 - LT(p)a 8 (T(s-^1 p))

pEr pEr

= LT(p)(T(p)-a 8 (T(s-^1 p)))

pEr

= (T, T - s *a T).

Hence the desired inequality follows from the Cauchy-Schwarz inequality,
since JJTll2 = 1. D

Here is the analogue of Lemma 4.2.3 for crossed products by am~nable
actions.

Lemma 4.3.3. Let A be a unital I'-C -algebra and T: r --- A be a finitely
supported function with support F, such that 0::::; T(g) E Z(A) for all g Er
andL 9 T(g)^2 = lA. Then, there existu.c.p. mapsr.p: A><la,rI'--- A®MF(C)
and 'ljJ: A® MF(<C)--- A ><la,r r such that for alls Er and a EA,
'ljJ o r.p(a>. 8 ) = (T a T(s))a>. 8 •

Proof. We already have a u.c.p. compression map r.p: A><la,rI'----"* A®MF(C)
such that

r.p(a>.s) = L a;^1 (a) ® ep,rlp EA® MF(C).

pEFnsF

4This definition comes from the characterization of (classical) amenability in terms of weak

containment of the trivial representation in the left regular representation.