The reduced C∗-algebra of the interior of the isotropy in any Hausdorff
\'etale groupoid G embeds as a C∗-subalgebra M of the reduced
C∗-algebra of G. We prove that the set of pure states of M with unique
extension is dense, and deduce that any representation of the reduced
C∗-algebra of G that is injective on M is faithful. We prove that there
is a conditional expectation from the reduced C∗-algebra of G onto M if
and only if the interior of the isotropy in G is closed. Using this, we prove
that when the interior of the isotropy is abelian and closed, M is a Cartan
subalgebra. We prove that for a large class of groupoids G with abelian
isotropy---including all Deaconu--Renault groupoids associated to discrete
abelian groups---M is a maximal abelian subalgebra. In the specific case of
k-graph groupoids, we deduce that M is always maximal abelian, but show by
example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for
pointing out the error); v2 shows there is a conditional expectation onto M
iff the interior of the isotropy is closed. v3: Material (including some
theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This
version published in Integral Equations and Operator Theor