- Coarse metric spaces 195
of finite symmetric subsets of r such that LJ En = r and EnEm c En+m for
every n, m. It is not hard to see that the function d on r x r defined by
d(s, t) = min{n: sC^1 E En}
is a proper right invariant metric on r. The second assertion is a nice
exercise. D
Let BR(x) = {y E X : d(y, x) < R} be the ball with center x and
radius R. Recall that a (discrete) metric space (X, d) has bounded geometry
if SUPxEX JBR(x)J < oo for every R > 0.
Definition 5.5.3. Let (X, d) be a metric space with bounded geometry. For
every S > 0, we set Ts(X) = {(x, y) EX x X : d(x, y) < S}. An operator
a E JB( .e^2 ( X)) is said to have finite propagation if the kernel of a is supported
on Ts(X) for some S (i.e., (a6y, Dx) =/= 0 only if d(x, y) < S). The translation
algebra A(X) is the *-algebra of all operators with finite propagation. The
closure of the translation algebra A(X) in JB(.e^2 (X)) is called the uniform
Roe algebra and is denoted by C~(X). Clearly, another metric d' on X which
is coarsely equivalent to d gives rise to the same algebras A(X) and C~(X).
Remark 5.5.4. If r is a countable group, C~(r) is easily seen to be the
same as the Roe algebra defined in Section 5.1 (which Proposition 5.1.3
identifies with .e^00 (r) ><lr r).
There is a notion of amenability for a coarse space defined via F¢lner
sets, and a coarse space X with bounded geometry is amenable if and only
if C~(X) has a tracial state. However, since every coarse metric space is
embeddable into an amenable coarse metric space, amenability does not say
much about the total structure of the space (unless the space is "coarsely
homogeneous"). But, there is an important invariant, defined by Yu, which
does pass to subspaces.
Definition 5.5.5 (Property A). We say a metric space (X, d) has property
A if for any R > 0 and c; > 0, there exist a Hilbert space 1-i, a map ~: X --+ 1-i
and a number S such that
(1) Jl~xll = 1 for every x EX,
(2) if d(x, y) < R, then Jl~x-:-~yJJ < E,
(3) and if d(x, y) ~ S, then (~y, ~x) = 0.Just as for groups, we say a bounded kernel k: X x X --+ C is positive
definite if the matrix [k(x, y)]x,yE%' is positive for every finite set i C X.
The following lemma is very similar to our previous work on groups.
Lemma 5.5.6. Let (X, d) be a discrete metric space with bounded geometry.
The following are equivalent: