5.6. Groupoids 207is algebraically positive, where gp("!) = (1 - lf(r('Y))l^2 )^112 gp("f). Take an
element e E Cc(G(o)) such that 0 ~ e ~ 1 and e(s('Y)) = 1 = e(r("!)) for
every 'YE supp h. Then, mh(e f e) = mh(f) for every f E Cc(G). Now,
it is not too hard to see that Stinespring's dilation of mh: Cc( G) ~ C ( G)
gives rise to a continuous -homomorphism on C(G), and hence mh is a
c.c.p. map on C*(G). (Let e play the role of the identity 1.) D
Corollary 5.6.17. Let G be an etale locally compact groupoid. If G is
amenable, then C* ( G) = C{ ( G) canonically.Proof. By Lemma 5.6.14, there exists a net (i E Cc( G) such that ll(ill£2(G) ~
1 and hi = (i (i ~ 1 uniformly on compact subsets of G. By Proposi-
tion 5.6.16, mhi is c.c.p. on C(G) and on C{(G). By the remarks following
the definition of the full groupoid C -algebra norm, we have mhi (a) ~ a
in C(G) for every a E Cc(G) and hence for every a E C(G). Moreover,
we observe that mhJa) E Cc(G) for every a E C(G). Now, we denote by
7f : C* ( G) ~ C{ ( G) the canonical quotient map and take a E ker 7r. Then,
we have (7r o mhJ(a) = (mhi o 7r)(a) = 0. Since 7r is injective on Cc(G), this
implies that mhi (a) = 0 for all i, and hence a = 0. D
At the time of this writing, the converse (whether C* ( G) = C{ ( G) im-
plies amenability of G) is not known.
Theorem 5.6.18. Let G be an etale locally compact groupoid. The following
are equivalent:
(1) G is amenable;
(2) there exists a net of positive-type functions hi E Cc(G) which con-
verges to 1 uniformly on compact subsets of G;
(3) C{(G) is nuclear.
Proof. (1) {::} (2): Leth E Cc(G) be a positive-type function and let E > 0.
Since >.(h) 2:: 0 in C{(G), there exists ( E Cc(G) such that 11>.(h-(**()ll < E.
By approximation, we are done.
(2) ==?-(3): Let A be a C*-algebra and Q: C{(G) ®max A~ C{(G) ®A
be the quotient map. We will prove that Q is injective. We first claim that
for every compact subset Kc G, there exists a constant CK> 0 such that
n n
II LA(jj) ® ajllc~(G)0maxA ~CK sup II Lfj('Y)ajll
j=l ryEK j=l
for every aj EA and fj E Cc(G) with supp fj CK. Since K is compact, by
a partition of unity reduction argument, we may assume that both sand r
are homeomorphisms on K. It follows that supp(ft * fj) C Q(O) for every i, j.