1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.4. Another approximation property 373

By Lemma D. 7, if <{Ji E qr] converges to 1 weak*, then m'Pi © idlB(£2) -t

idc~(r)GlB(£2) in the point-weak topology. Hence, by the Hahn-Banach The-

orem, there is a net of convex combinations of the m'Pi 's which converges to

idq (r) in the strong stable point-norm topology. D

It follows from this result that a nonexact group does not have the AP.

The algebraic direct product of infinitely many copies of (Z/2Z) 2 lF2 is not

weakly amenable (Corollary 12.3.7 and Theorem 12.3.13), but it has the

Haagerup property (Theorem 12.2.11) and the AP.

Proposition 12.4.10. The class of groups with the AP is closed under

extensions.

Proof. Let {1} -t A -t r ~ r -t {1} be a short exact sequence of groups
with A and r having the AP. We denote by Z the weak -closure of qr] in
B 2 (r). It suffices to show 1 E Z.
Consider the inclusion 1,,: B2(A) <----t B2(r), where i(<p)(s) = <p(s) for
s E A, and 1,,( <p) = 0 off A. It is clear that 1,, is an isometry such that
1,,(qA]) c qr]. We claim that 1,, is weak-continuous. For this, it suffices to
show 1,,(Q(I')) c Q(A), which in turn is equivalent to the trivial assertion
that 1,,(b"s) E Q(A) for every s Er. Since A has the AP, we have XA = i(l) E
Z. It also follows that XsA E Z for every s E I'. Therefore, for the inclusion
1f: B 2 (r) <----t B 2 (r) given by 7r(<p) = <po1f, we have 7r(qr]) c Z. Since 1f
is a weak* -continuous contraction (why?), the assertion 1 E Z follows from
the AP of r. D

Exercises

Exercise 12.4.1. Check the details of the proof of (2) =?- (1) in Theo-
rem 12.4.9.

Exercise 12.4.2. Let C~(r) = C*(.A.(r),f<Xl(r)) be the uniform Roe alge-
bra (Section 5.1). We say the group r satisfies the invariant translation
approximation property (ITAP) if

c~(r) n L(r) = C~(r).

Prove that if r has the AP, then r has the ITAP. (At present, it isn't known
whether there exists a group without the ITAP - cf. [167, 199].)

Exercise 12.4.3. Let r be a group and A c r be a co-amenable subgroup.
Prove that r has the AP if A has the AP.

Exercise 12.4.4. Let r be a group acting on a locally finite tree. Prove
that r has the AP if one of the vertex stabilizers has the AP.