5.6. Groupoids 203The topology of Gu is the relative product topology, where I is discrete.
The groupoid Gu is an etale locally compact groupoid if G is.Definition 5.6.9. Two etale locally compact groupoids G and G' are said
to be Morita equivalent if there are localizations Gu and G~,, respectively,
which are isomorphic as topological groupoids. (NB: This definition is
adapted for etale groupoids.)
Example 5.6.10 (Groupoids and coarse spaces). To a coarse space X, one
can associate a translation groupoid G(X) (roughly, a groupoid of partial
bijections on X). The reduced C-algebra of G(X) turns out to be iso-
morphic to the uniform Roe algebra C~(X). See [167, Chapter 10] for the
construction of G(X) and the C-isomorphism result.
Example 5.6.11 (Graph groupoids). There is a way to associate a groupoid
to a directed graph, but the construction is technical and authors grow tired;
see [112, 140].
Now we turn to the construction of C* -algebras associated to groupoids.
Let G be an etale locally compact groupoid. Thus,
Gx ={'I' E G: s('Y) = x} and ax = {'y E G : r('Y) = x}
are discrete in G for every x E Q(O). The groupoid algebra Cc( G) is the
-algebra of all continuous compactly supported functions f: G ----+ CC with
composition and -operation given by
(f*g)('Y) = 2= f(a)g(f3) = 2= f('Yf3-^1 )g(f3) and f*('Y) = J('Y-^1 ).
cxf3='Y (3EG s('y)If K c G is a compact subset, we can cover it with finitely many open subsets
on which s is a homeomorphism, and hence we have supx \Kn Gx\ < oo.
It follows that f * g indeed belongs to Cc(G). For the convenience of the
reader, we note that
(f* * g)('Y) = 2= f(f3)g(f3'f').
(3EGr(-y)The groupoid algebra Cc(G) is naturally a right Co(GC^0 ))-module with right
action and Co(G(O))-valued inner product defined by
(e. !)(')') = eb)f(s('T')) and (e,rJ)(x) = 2= eb)rJ('/')
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