1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

198 5. Exact Groups and Related Topics

for x, z EX. Then, for some S > 0, we have supp (x c Bs(x) for all x EX.

Moreover, ll(xll = 1 for all x EX. Indeed,

ll(xll

2

= L II I:ez(c5z,azc5x)ll~

z l

=I: I: (6, emh-l(c5z, a1c5x)e2cx) (c5z, amc5x)e2cx)

l,m z

l,m

= (c5x, (f1/;, e'lf;)?.-l®G:(,(X)c5~)e2(x) = 1.

Finally, let (x,y) E TR(X) and choose v E :g so that vc5x = 6y. Since rp(v) is

a contraction in Mn ( C), we have

((y, (x)e^2 (X) = L II L6(c5z, azc5y) 117-lll L em(c5z, amc5x)ll7-l

z l m

2: L II L 6(c5z, azc5y) 117-lll ( r.p( v) 0 1) Lem (c5z, amc5x) 117-l

z l m

l,m z

= l(c5y, (e'lf;, (rp(v) 01)e'lf;)?-l&c::,cx)c5x)e2cx)I

= l(c5y, (1/1°r.p)(v)c5x)e 2 (x)IRj1.

This implies ll(y - (xii is small and hence X has property A. D

We finish off this section with two important facts. First, property

A implies uniform embeddability into Hilbert space - this has important

consequences for the Baum-Connes conjecture. The second follows from the

first: there aire coarse spaces which don't have property A.

Lemma 5.5.8. A metric space X with property A is coarsely isomorphic to

a subset of a Hilbert space.

Proof. For each n, we find en: X --+ Hn and numbers Sn satisfying Def-
inition 5.5.5 for R = n and c = 2-n. Define 1-{ = EB Hn and f: X --+ 1-{
by

where o EX is a fixed base point. It follows that f(o) = 0 and

111(x) - J(y)ll :::; 2d(x, y) + I: 11e~ - e;11 :::; 2d(x, y) + i.

n?:,d(x,y)