1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

204 5. Exact Groups and Related Topics

fore, 'fJ E Cc(G) and f E Co(G(o)). Observe that (e, 'fJ) E Cc(G(o)) is the

restriction of e * 'fJ E Cc(G) to Q(O). Denote by L^2 (G) the Hilbert Co(G(o))-

module16 arising from the completion of Cc(G). The left regular representa-

tion>.: Cc(G)---+ lffi(L^2 (G)) is defined by

>.U)e = 1 * e

for J,e E Cc(G). It is not hard to see that 11>.(f)ll < oo and>.(!)*=>.(!*).

The reduced groupoid C* -algebra C~ ( G) is the norm closure of >.(Cc( G)) c

lffi(L^2 (G)). Since Q(O) is clopen in G, we have Cc(G(o)) C Cc(G) and the

restriction map Cc(G) ---+ Cc(G(o)) gives rise to a conditional expectation

E from C~(G) onto Co(G(o)) C C~(G), which implements the isomorphism

L^2 (G) ~ L^2 (C~(G),E) (see Section 4.6). For every x E Q(o), we define a

*-representation Ax of Cc(G) on f^2 (Gx) by

(>.x(J)e)(ry) = 2: Jh/3-^1 )e(/3).

(3EGx

Lemma 5.6.12. For f E Cc(G), let

llfll1,s = sup L lf(ry)I and llfll1 = max{llfll1,s, llf*lh,s}·

. xEG(O) 'YEGx
Then, for every f E Cc(G), we have
llflloo S 11>.(f)ll = sup 11>.x(f)ll S llfll1.
xEG(O)

Proof. We only prove supxEG(o) 11>.x(f)// S llfll1; the rest is trivial. For

f E Cc(G) and e, 'f] E f^2 (Gx), we have

1\e,>.xU)'fJW =I I: e(a)f(af3-^1 )'fJ(f3)l

2

S llell

2

llf*ll1,sllfll1,sll'fJll

2

·

Hence 11>.x(f)ll S llf*11Vs

2

llflli;s

2

S llflll·

a,(3EGx

D

The full groupoid C -algebra C ( G) of the etale locally compact groupoid
G is defined as the norm-completion of Cc( G) with respect to the norm

llfllo(G) =sup llK(f)ll,
where the supremum is taken over all (cyclic) -representations 7l" of Cc(G)
on Hilbert spaces, which are bounded on the commutative *-subalgebra.
16 If one wants an ordinary Hilbert space, choose a regular Borel measure μ on Q(O) with
full support and set (~,'T/)μ = fa(O)(~,ry)(x)dμ(x). Then we obtain a Hilbert space L^2 (G,μ) ~
£2(G) ®ca(G(o)) L2(Q(Dl,μ).