1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.6. Groupoids 205

Cc(G(o)). Let U c G be an open subset on which both s and r are
homeomorphisms. Then, for every f E Cc( G) with supp f c U, we have
f f E Cc(GCO)) with (f f)(s("!)) = lf('Y)l^2 for 1 E U. It follows that
llfllc(G) = llfll~ for such f. Thus, by a partition of unity argument, for
every compact subset K C G, there exists a constant CK > 0 such that
llgllc(G) :::; CKll9lloo for every g E C(K) = {g E Cc(G) : suppg CK}. In
particular, llgllc(G) < oo for any g E Cc(G).^17 In passing, we observe that
C(K) is closed in C(G). (Do you see the point?)
Let G = X ><1 r be a transformation groupoid. We leave it to the reader
to check that C~(G) ~ C(X) ><Irr and C*(G) ~ C(X) ><1 r. It is a good
exercise to check that the action is amenable if and only if there exists
a net of compactly supported nonnegative functions μi : G --+ C such that
2.:,t μi(x, t)--+ 1and2.:,t lμi(x, t)-μi(sx, st)I--+ 0 for (x, s) E G uniformly on
compact subsets of G. Noticing that (sx, st)= (sx, s)(x, t) in G, we are led
to the definition of amenability for general etale locally compact groupoids
(though multiplication on the left gets moved to the right for a technical
reason).
Definition 5.6.13. An etale locally compact groupoid G is said to be
amenable if there exists a net of compactly supported nonnegative func-
tions μi : G --+ C such that
and
(3EGr(7) (3EGr(7)

for 1 E G, uniformly on compact subsets of G.

As usual, we have an £^2 -characterization of amenability.

Lemma 5.6.14. An etale locally compact groupoid G is amenable if and only

if there exists a net (i E Cc(G) such that ll(illL2(G) :::; 1 and ((i * (i)('Y)--+ 1,

for 1 E G, uniformly on compact subsets of G.

Proof. First we'll prove the "only if'' direction. Let μi be as in the definition

of amenability and set fi(x) =max{ (μil^2 , μi1^2 )(x), 1}. Then the functions

(i('Y) = μi("/)^112 fi(s("!))-^1!^2 satisfy the desired properties. For the "if''

direction, define μi('Y) = l(i('Y)l^2 and do a computation. D

The functions (i * (i are examples of positive-type functions.

Definition 5.6.15. A function h: G --+ C on an etale locally compact

groupoid G is said to be of positive type if [h( a,B-^1 )]a,(3E~ is positive definite

for every x E Q(O) and every finite subset~ C Gx.

17 we actually have ll91lo*(G) :::; llgllr, where ll9llr is defined in Lemma 5.6.12. See page 43

in [140].