Some results regarding the ideal structure of C∗$C^*$ ‐algebras of étale groupoids

We prove a sandwiching lemma for inner‐exact locally compact Hausdorff étale groupoids. Our lemma says that every ideal of the reduced C∗$C^*$ ‐algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced C∗$C^*$ ‐algebra, and triples consisting of two nested open invariant sets and an ideal in the C∗$C^*$ ‐algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced C∗$C^*$ ‐algebras of inner‐exact locally compact Hausdorff étale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.


Introduction
The purpose of this paper is to investigate the ideal structure of the reduced C *algebras of locally compact Hausdorff étale groupoids.This very broad class of C * -algebras contains all reduced crossed products of commutative C * -algebras by discrete groups.It also includes graph C * -algebras [KPRR97], higher-rank graph C * -algebras [KP00], the models described by Spielberg [Sp07] and Katsura [Ka2008] for Kirchberg algebras, the stable and unstable Ruelle algebras of Smale spaces (up to Morita equivalence), and many self-similar action C * -algebras [EP2017].
Among the more natural invariants of a C * -algebra, but also among the most difficult to compute, is its lattice of ideals.In the situation of étale groupoid C * -algebras, definitive theorems are available for C * -algebras of amenable groupoids that are essentially principal in the sense of Renault [Re91], graph C * -algebras [aHR97,HS04], and for C * -algebras of a single local homeomorphisms [Ka2021], but few other truly general results about ideal structure of groupoid C * -algebras are available.
The analysis of ideals in étale groupoid C * -algebras typically has two components.The first is concerned with what we call here dynamical ideals, and is well understood.The continuous functions on the unit space G (0) of an étale groupoid G embed as a σ-unital subalgebra D of the groupoid C * -algebra.So each ideal I of C * (G) yields an ideal I ∩ D of D and hence an open subset U I of G (0) on which it is supported.This U I is invariant in the sense that if s(γ) ∈ U I then r(γ) ∈ U I .If I is generated as an ideal by I ∩ D, we call it a dynamical ideal.The assignment I → U I is a lattice isomorphism between dynamical ideals of C * (G) and open invariant sets of G (0) , giving a complete description of the dynamical ideals.In particular, by identifying the essentially principle (now sometimes referred to instead as strongly effective) and amenable groupoids for which every ideal of C * r (G) is dynamical, Renault gives a complete description of the ideal structure for this class of groupoid C * -algebras [Re91].
The second component of the analysis is more complicated.It amounts to understanding the collection of all possible ideals that have fixed intersection with D. For full C * -algebras, this is, in general, hopelessly intractable: there is a zoo of ideals contained in the kernel of the regular representation, which has trivial intersection with D, alone.So we are led to restrict our attention to reduced C * -algebras.Another problem arises almost immediately: given an open invariant set U, the restriction map f → f | G\G| U on C c (G) extends to a homomorphism C * r (G) → C * r (G \ G| U ) whose kernel contains the dynamical ideal I U associated to U. In the setting of full C * -algebras, this containment is an equality, but for reduced C * -algebras it need not be: as Willet's example [Wi15] shows, the quotient C * r (G)/I U can coincide with the full C * -algebra C * (G \ G| U ), and we encounter the same zoo of ideals as before.So we restrict attention further to groupoids that are inner-exact in the sense that C * r (G)/I U ∼ = C * r (G\G| U ) for every open invariant set U. Lest this seem overly restrictive, note that this includes all amenable étale groupoids G, and therefore all nuclear étale groupoid C * -algebras [A-DR00].
In this setting, existing results rely, explicitly or otherwise, on a kind of sandwiching lemma.This technique was developed by an Huef and Raeburn [aHR97] to analyse Cuntz-Krieger algebras.Here the dynamical ideals are better known as gauge-invariant ideals (see Proposition 3.9).To understand the ideals of a Cuntz-Krieger algebra, an Huef and Raeburn concentrate on primitive ideals and demonstrate that for each primitive ideal I there are a unique smallest gauge-invariant ideal K containing I and largest gaugeinvariant ideal J contained in I.They then analyse the quotient K/J, which is itself Morita equivalent to a graph algebra-but of a graph consisting of just one vertex and one edge.The C * -algebra of this graph is C(T), so its ideal structure is well understood, and their analysis proceeds from there.A similar idea was used in [HS04], and again in [Ka2021] to understand ideal structure first for graph C * -algebras and then for topologicalgraph algebras, viewed as C * -algebras associated to singly generated irreversible dynamics.
Another instance of the same idea appears in Ara and Lolk's very interesting work on partial actions [AL18].They identify a relative strong topological freeness condition that generalises Renault's topologically principle condition in the setting of transformation groupoids for partial actions.They show that relative strong topological freeness guarantees the existence of an obstruction ideal : a smallest dynamical ideal of C * r (G) that contains every ideal with trivial intersection with D. This can again be regarded as a kind of sandwiching result, but with the quantifiers switched: there exists a pair of dynamical ideals, namely the zero ideal and the obstruction ideal, that sandwich every ideal that has trivial intersection with D. One of our motivations in writing this paper is that, because this particular aspect of Ara and Lolk's paper appears as a technical step along the way to their main objective, it is in danger of receiving less attention than we think it deserves, and we want to advertise the idea more broadly.
In this paper, we take up the idea of the sandwiching lemma and of Ara and Lolk's relative strong topological freeness condition and obstruction ideal.We first establish a general sandwiching lemma for groupoid C * -algebras (Lemma 3.4): given any innerexact locally compact Hausdorff étale groupoid G, and any ideal I of C * r (G), there are a unique smallest dynamical ideal K containing I and largest dynamical ideal contained in I.As a result the ideals of C * r (G) are parameterised by triples (U, V, J) consisting of open invariant sets U ⊆ V ⊆ G (0) , and an ideal J of C * r (G| V \ G| U ) that has trivial intersection with D and vanishes nowhere on G| V \ G| U (Theorem 3.7).
We then adapt Ara and Lolk's notions of topological freeness and strong topological freeness at a point (see also Renault's notion of discretely trivial isotropy [Re91]), and of relative strong topological freeness, from their setting of partial actions of groups to the setting of étale groupoids.We identify a condition on étale groupoids, which we phrase as being jointly effective where they are effective, that ensures that C * r (G) admits an obstruction ideal in the sense of Ara and Lolk (see Theorem 4.12 and Corollary 4.14).We also show that this obstruction ideal is minimal in the strong sense that there exists an ideal that has trivial intersection with D and whose support exhausts the support of the obstruction ideal.We show that any groupoid whose isotropy groups are all either trivial or infinite cyclic is jointly effective where it is effective.This includes all graph groupoids and groupoids arising from single local homeomorphisms.In our companion paper [BCS23], we show how to use our results to give a complete description of the ideal structure of a large class of Deaconu-Renault groupoid C * -algebras, including those considered in [aHR97, HS04, Ka2021] and all C * -algebras of rank-2 graphs.
The paper is arranged as follows.We introduce the background we need in Section 2. In Section 3 we prove our sandwiching lemma and explore its consequences.In Section 4 we introduce the notions of a groupoid being effective at a unit, strongly effective at a unit, and being strongly effective where it is effective.We then prove that such groupoids admit an obstruction ideal, and discuss some examples.Finally in Section 5, we present examples of groupoids that are jointly effective where they are effective, and describe the support of the obstruction ideal; in particular, we devote Section 5.2 to showing exactly how our work in Section 4 generalises the ideas of Ara and Lolk.

Preliminaries
2.1.Hausdorff étale groupoids.We will always be working with topological groupoids that are locally compact, Hausdorff, and étale, and we shall adopt most of the notation and terminology from [Si20] (see also [Re80]).
We consider the unit space G (0) as a locally compact Hausdorff subspace of G, and we denote range and source maps by r, s : G → G (0) .A bisection is a subset B of G such that both r and s restrict to injective maps on B. That G is Hausdorff means that the unit space G (0) is a closed subset of G, and that G is étale (in the sense that the range and source maps are local homeomorphisms) implies that G (0) is also open, that G has a basis consisting of open bisections, and that the range and source fibres over a unit x ∈ G (0)  given by G x = {γ ∈ G : r(γ) = x} and G x = {γ ∈ G : s(γ) = x}, respectively, are discrete in the relative topology.In particular, the isotropy group over a unit x ∈ G (0) given as the intersection The isotropy subgroupoid is then the group bundle We write I • (G) for the topological interior of the isotropy of G.A Hausdorff groupoid G is said to be effective if I • (G) = G (0) , that is, the interior of the isotropy subgroupoid with the subspace topology coincides with the unit space.When G is second-countable this coincides (using a Baire category argument) with the notion of G being topologically principal in the sense that G (0) has a dense set of points with trivial isotropy.2.2.Reduced groupoid C*-algebra.We will be working with the reduced groupoid C * -algebras of locally compact Hausdorff étale groupoids.We follow the exposition of [Si20].
The convolution algebra C c (G) of a locally compact Hausdorff étale groupoid G is the set of compactly supported and complex-valued functions on G equipped with the convolution product for all f, g ∈ C c (G) and γ ∈ G, and the involution f given by is both open and closed in G, the commutative algebra C 0 (G (0) ) sits naturally as a subalgebra of C * r (G), and we refer to C 0 (G) as the diagonal subalgebra.Note that C 0 (G (0) ) need not be a C * -diagonal (in the sense of Kumjian [Ku86]) nor a Cartan subalgebra (in the sense of Renault [Re08]).
Renault [Re80, Proposition II.4.2] shows (Renault makes the standing assumption that the groupoids considered there are second-countable, but that assumption is not needed for the following) that any element in the reduced groupoid C * -algebra may be thought of as a function on the groupoid.More precisely, there exists a linear and norm-decreasing map j : for all a ∈ C * r (G) and γ ∈ G, and j is the identity on C c (G).The reduced groupoid C *algebra admits a faithful conditional expectation E : C * r (G) → C 0 (G (0) ) onto the diagonal given by restriction of functions in the sense that j(E(a)) = j(a)| G (0) for all a ∈ C * r (G) [Si20, Proposition 10.2.6].Renault shows that for a, b ∈ C * r (G), the convolution formula for j(a) * j(b) is a convergent series that converges to j(a * b).
A subset U of G (0) is G-invariant (or simply invariant) if r(GU) ⊆ U, and the reduction of an open subgroupoid of G (and hence locally compact, Hausdorff, and étale), and the inclusion i . This is an ideal with the property that I U ∩ C 0 (G (0) ) = C 0 (U) and I U is generated as an ideal by C 0 (U).We shall refer to such ideals as dynamical ideals (see Definition 3.1).
The complement G (0) \U is a closed invariant set of units, and there is a * -homomorphism The groupoid ) is exact, (see [A-D19, Definition 3.7] and also [BL17, Definition 3.5]).Any amenable groupoid is inner exact.Combining Proposition 4.23 and Theorem 7.10 in [A-D21], we also see that the (partial) crossed product groupoid of an exact group acting (partially) on a second-countable locally compact Hausdorff space is inner-exact.Willett's example of a nonamenable groupoid whose full and reduced C * -algebras coincide is not innerexact [Wi15].Remark 2.2.The empty set satisfies the axioms defining a locally compact Hausdorff étale groupoid.By convention, we take the C * -algebra of the empty groupoid to be the zero C * -algebra; in particular (2.1) collapses to the exact sequence }.We thank the referee for pressing us on this point.

A sandwiching lemma for Hausdorff étale groupoids
The characterisations of the primitive-ideal spaces of graph C * -algebras of [HS04] and [aHR97] were founded on the "sandwiching lemmas" [HS04, Lemma 2.6] and [aHR97, Lemma 4.5] that show that every primitive ideal is sandwiched between a pair of uniquely determined gauge-invariant ideals.Here we observe that a similar sandwiching lemma holds for ideals of reduced Hausdorff étale groupoid C * -algebras.Definition 3.1.We say that an ideal I in a reduced groupoid C * -algebra C * r (G) is dynamical if it is generated as an ideal by its intersection with the diagonal subalgebra that is both a dynamical ideal and a purely non-dynamical ideal.Though linguistically unsatisfactory, this convention simplifies the statements of our key results: in Proposition 3.3 treating {0} as a dynamical ideal avoids treating the open invariant set ∅ as a special case; but later in Theorem 3.7, treating {0} as a purely non-dynamical ideal avoids treating dynamical ideals as a special case-see Remark 3.8.
In the context of Deaconu-Renault groupoids, the dynamical ideals are precisely the usual gauge-invariant ideals-see Proposition 3.9.Proposition 3.3.Let G be a locally compact Hausdorff étale groupoid.The map U → I U is a lattice isomorphism from the lattice of open invariant subsets of X to the lattice of dynamical ideals of C * r (G).For each open invariant U ⊆ G (0) , we have Proof.The map U → I U is always an injection [Si20, Theorem 10.3.3], and surjectivity follows from the definition of dynamical ideals.Proposition 10.3.2 of [Si20] shows that I U is the closure of C c (G| U ) ⊆ C c (G).In particular, supp(I U ) ⊆ G| U , and Since lattice isomorphisms preserve least upper bounds and greatest lower bounds, it follows from Proposition 3.3 that, for example, I U ∩ I V = I U ∩V and I U + I V = I U ∪V for all open invariant U and V .
We now state our sandwiching lemma.
Lemma 3.4 (The sandwiching lemma).Let G be a locally compact Hausdorff étale groupoid that is inner-exact and let I be an ideal of Consider the open and invariant subsets For each x ∈ U, choose f x ∈ I ∩ C 0 (G (0) ) such that f x (x) = 0. Then {f x : x ∈ U} generates C 0 (U) as an ideal of C 0 (G (0) ), and it is contained in I. Hence I U ⊆ I. Suppose that U ′ is an open subset of G (0) strictly containing U and fix x ∈ U ′ \ U and f ∈ C c (U ′ ) with f (x) = 0. Then f ∈ I U ′ but f ∈ I U by definition of I U .In particular, f / ∈ I, so I U ′ ⊆ I.This proves that I U is the largest dynamical ideal contained in I.
The set V is open because j(a) is continuous for every a ∈ C * r (G).We claim that V = s(supp(I)).That V ⊆ s(supp(I)) is obvious.For the reverse inclusion, suppose that a ∈ I and j(a)(γ) = 0.For any open bisection B containing γ −1 and any ) be the faithful conditional expectation onto the diagonal and observe that E(I) ⊆ C 0 (V ).Since G is inner-exact, it follows from [BL17, Lemma 3.6] that I is contained in the ideal in C * r (G) generated by E(I), so we find that I ⊆ I V as wanted.To see that V is minimal with this property, suppose that Remark 3.5.If the ideal I in Lemma 3.4 is a purely non-dynamical ideal of C * r (G), then U is empty, and then I U = {0}; if I is a dynamical ideal, then V = U and I U = I.

Consider a pair of nested open invariant subsets
extending the canonical inclusion of algebras of compactly supported functions.For these maps, the diagram Lemma 3.6.Let G be a locally compact Hausdorff étale groupoid that is inner-exact.
Let I be an ideal of C * r (G) and let U and V be the open invariant sets of Lemma 3.4.
) that is purely non-dynamical and has full support.
Next we show that J has full support.Clearly, supp(J) = 0, and we conclude that supp(J) = G| V \U .
Let T (G) be the collection of triples (U, V, J) where U ⊆ V ⊆ G (0) are nested open and invariant subsets and J is a purely non-dynamical ideal in C * r (G| V \U ) with full support.Theorem 3.7.Let G be a locally compact Hausdorff étale groupoid that is inner-exact.There is a bijection Θ from T (G) to the collection of ideals of to the triple (U, V, J) ∈ T (G) consisting of the sandwich sets U ⊆ V and the purely non-dynamical ideal J ⊳ C * r (G| V \U ) with full support of Lemma 3.4.Remark 3.8.It is important in the statement of Theorem 3.7 that ∅ is a groupoid, that its reduced C * -algebra is {0}, and that {0} is a purely non-dynamical ideal of C * r (G): the dynamical ideals of C * r (G) are in the range of Θ because each I U = Θ(U, U, {0}).Proof of Theorem 3.7.The map Θ takes values in the ideals of C * r (G) by definition.To see that Θ is injective, fix (U, V, J) ∈ T (G) and let I = Θ(U, V, J).We will prove that U and V are the sandwiching sets U I , V I obtained from Lemma 3.4 applied to I, and that J = ι −1 V \U (π U (I)).This defines a left inverse to Θ, defined on the image of Θ, which implies that Θ is injective.
We have and the latter has trivial intersection with G (0) \ U (since J is purely non-dynamical).
As π U implements restriction of functions, we see that ). Observe that E(Θ(U, V, J)) ⊆ C 0 (V ).By [BL17, Lemma 3.6], Θ(U, V, J) is contained in the ideal generated by E(Θ(U, V, J)), so we see that Θ(U, V, J) ⊆ I V .In particular, supp(Θ(U, V, J)) ⊆ supp(I V ) = G| V .On the other hand, since supp(J) = G| V \U we have G| V which contradicts our observation above.Therefore, V is the smallest such open invariant subset.Finally, observe that = J, and this completes the proof that Θ is injective.
To see that it is surjective, fix an ideal ).We claim that K := Θ(U, V, J) is equal to I, which will establish surjectivity of Θ.By definition, both I and K are ideals of C * r (G) that contain I U , so it suffices to show that To link Lemma 3.4 back to the results [HS04, Lemma 2.6] and [aHR97, Lemma 4.5] that inspired it, we observe that for Deaconu-Renault groupoids, the dynamical ideals employed above are precisely the gauge-invariant ideals of the C * -algebra of a Deaconu-Renault groupoid.The result is certainly well-known, but we are not aware that it has been recorded explicitly elsewhere in this generality.For the case of finitely aligned higher-rank graphs, this was observed in [Li21, Lemma 7.5].
Recall that if T : Proof.Since γ z (f ) = f for all z ∈ T k and f ∈ C 0 (G (0) ), the ideals of C * (G T ) generated by subsets of C 0 (G (0) ) are gauge invariant.In particular, each I U is gauge-invariant.
The map U → I U is an injection [Si20, Theorem 10.3.3].For surjectivity, we follow the second paragraph of the proof of [Si20, Theorem 10.3.3],dropping the assumption that G is strongly effective but fixing a gauge-invariant ideal I, until its penultimate sentence.At that point, while GW need not be effective, we observe that GW is identical to the groupoid of the topological higher-rank graph Λ defined by Λ n = X × {n} for all n, whose range and source maps are given by s(x, n) = (T n (x), 0) and r(x, n) = (x, 0) and with the factorisation rules (x, m)(T m (x), n) = (x, m + n) = (x, n)(T n (x), m).We may now apply the gauge-invariant uniqueness theorem of [CLSV11, Corollary 5.21] in place of [Si20, Theorem 10.3.3] to see that π is injective, and the surjectivity of U → I U follows.The final statement follows from Proposition 3.3.

Effectiveness at a unit and the obstruction ideal
In this section we introduce the notions of effectiveness at a unit and joint effectiveness at a unit for étale groupoids.The key property that emerges is that of being jointly effective where effective.This is inspired by the notions in [AL18, Section 7] of (strong) topological freeness at a point for a partial group action.The points in the unit space of a groupoid that are not effective comprise an open invariant set and hence a dynamical ideal that we call the obstruction ideal.Our main results in this section (Theorem 4.12 and Corollary 4.14) say that if a Hausdorff étale groupoid is inner-exact and its full and reduced C * -algebras coincide (Anantharaman-Delaroche calls this the weak containment property [A-D19]), then the obstruction ideal contains all purely non-dynamical ideals, and is minimal with this property.
Recall that a groupoid G is effective if the interior I • (G) of the isotropy is equal to the unit space G (0) .For x ∈ G (0) , we write I • (G) x for the intersection of G x with I • (G).Definition 4.1.A locally compact Hausdorff étale groupoid G is effective at a unit x ∈ G (0) if I • (G) x = {x}.Equivalently, G is effective at x if for any nontrivial isotropy element γ ∈ I(G) x \ {x} and any open bisection B in G \ G (0) containing γ there exists y ∈ s(B) such that r(By) = y.When the groupoid is understood, we may just say that the unit is effective.We let G (0) eff denote the collection of effective units.Any unit with trivial isotropy is effective.An isolated unit with nontrivial isotropy is not effective.
We have the following general description of the units that are not effective.This also shows that our terminology is consistent with the literature on effective groupoids.Lemma 4.2.Let G be a locally compact Hausdorff étale groupoid.We have and this is an open and invariant subset of G (0) .Consequently, G eff is closed and invariant.Moreover, G is effective if and only if G is effective at each of its units.
Proof.Suppose that G is not effective at x ∈ G (0) .Then x has nontrivial isotropy and any is an open neighbourhood of x consisting of points that are not effective, so In order to see invariance, let x ∈ s(I • (G) \ G (0) ) and take γ ∈ G with x = s(γ) and r(γ) = z = x.We will show that z is not effective.
γ is an open bisection containing γηγ −1 (which is isotropy over z), and it consists only of isotropy elements, because B η consists only of isotropy elements.Therefore, The final statement is a direct consequence of (4.1).
The obstruction ideal defined below will play a central role in Theorem 4.12.
Definition 4.3.Let G be a locally compact Hausdorff étale groupoid.The set of all units that are not effective is an open and invariant subset of G (0) , so it determines a dynamical ideal I G (0) \G (0) eff of C * r (G).We call this the obstruction ideal and denote it by J ob .This terminology is explained in Remark 4.16.
We let G eff denote the reduction of G to the closed invariant subset of effective points.The unit space of G eff then coincides with G (0) eff .We require a groupoid analogue of the notion of strong topological freeness introduced in [AL18, Section 7].
Definition 4.4.A locally compact Hausdorff groupoid G is jointly effective at a unit x ∈ G (0) if for any finite collection of nontrivial isotropy elements γ 1 , . . ., γ n ∈ I(G) x \ {x} and any open bisections B 1 , . . ., B n in G\G (0) such that γ i ∈ B i there exists y ∈ n i=1 s(B i ) such that r(B i y) = y for all i = 1, . . ., n.
Remark 4.5.If G is effective, then it is jointly effective at every unit.More generally, any unit in an open set of effective points is jointly effective.
For the first assertion, first suppose that G is effective, and fix x ∈ G (0) and γ 1 , . . ., γ n ∈ I(G) x \ {x}.Fix open bisections B i in G \ G (0) containing γ i .By shrinking if necessary, we can assume that W := s(B i ) = s(B j ) for all i, j.Since G is effective, each B i ∩ I(G) has empty interior.So for each i, the set ) is open and dense in W . Hence i W i is open and dense, and in particular nonempty.Now any y ∈ i W i satisfies r(B i y) = y for all i.
For the second assertion, suppose only that U is an open subset of G (0) contained in G eff .The first assertion applied to G| V shows that x is jointly effective in G| V , and hence in G.
It is possible for a groupoid to be effective at a unit but not jointly effective at that unit-see Example 4.8.This leads us to an analogue of Ara and Lolk's notion of relative strong topological freeness.
Definition 4.6.Let G be a locally compact Hausdorff étale groupoid.We say that G is jointly effective where it is effective if G is jointly effective at every point in G (1) By Remark 4.5, if G is effective then it is jointly effective where it is effective.In particular, if G is principal, then it is jointly effective where it is effective.
(2) Suppose G is a Hausdorff étale group bundle (for example, G is a nontrivial discrete group).Since Since G is trivially effective at x when G x x = {x}, it follows that G is jointly effective where it is effective.We have G (0) eff = {x : G x = {x}}, and the obstruction ideal is generated by C 0 ({x : G x = {x}}).
(3) In particular, Willett's groupoid [Wi15] consists entirely of isotropy, and hence is jointly effective where it is effective.It is not inner-exact.The obstruction ideal is the whole reduced groupoid C * -algebra.
The next examples show that groupoids need not be jointly effective where they are effective and that the property of being jointly effective where effective does not necessarily pass to reductions to closed invariant subsets.This latter permanence property does hold in groupoids all of whose nontrivial isotropy groups are infinite cyclic (see Section 5.1).
In this example, the interior of the isotropy and the only effective unit is (0, 0) ∈ X.
Every point in X has nontrivial isotropy (so G ϕ⊕ψ is not effective).More specifically, the isotropy group of every point that is not the origin is isomorphic to Z/2Z while the isotropy group at the origin is isomorphic to Z/2Z ⊕ Z/2Z.The origin is the only point that is effective, but it is not jointly effective.Therefore, G ϕ⊕ψ is not jointly effective where it is effective.
Ara and Lolk [AL18, Section 7] exhibit an example of a partial action that shows that their relative strong topological freeness is not automatic, and their example can be adapted to our groupoid setting.
Example 4.9.We can extend Exel's cross to see that being jointly effective where effective does not pass to closed invariant subgroupoids.To see this, let X be as in Exel's cross, and let Y = X × [−1, 1].
Neither a nor b fixes any point in Y \ X because both invert the t-coordinate.Since the only point in X fixed by ab is the point (0, 0) ∈ X, the only points in Y fixed by ab are those of the form ((0, 0), t).So G ϕ, ψ := Y ⋊ (Z/2Z) 2 is effective, and in particular jointly effective where it is effective.However, its reduction to the closed invariant set X is Exel's cross, which is not jointly effective where it is effective.
The next lemma is an easy adaptation of [Ex17, Lemma 29.4] from partial actions of groups to groupoids, so we give just a fairly succinct proof.
Lemma 4.10.Let G be a Hausdorff étale groupoid, let x ∈ G (0) be a unit, and let B be an open bisection such that B ∩ I(G) x = ∅.Let f ∈ C c (G) be such that f has support in B. Given ε > 0 there exists h ∈ C 0 (G (0) ) satisfying 0 h 1, h is constantly 1 on a neighbourhood of x, and hf h < ε.
The next two results say that when an inner-exact groupoid G whose full and reduced C * -algebras coincide is jointly effective where it is effective, its obstruction ideal J ob is the minimal dynamical ideal that contains all purely non-dynamical ideals of C * r (G).The proof of the first result closely follows that of [AL18, Theorem 7.12] (which does not require the weak containment property) with only minor modifications.
Remark 4.11.The hypothesis below that the sequence 0 for example, G is inner-exact (in particular, if it is amenable).However, it also holds trivially if G is effective, and we invoke it in that situation in Proposition 4.15.So we have stated Theorem 4.12 accordingly.Theorem 4.12.Let G be a locally compact Hausdorff étale groupoid that is jointly effective where it is effective.Let J ob be the obstruction ideal in C * r (G) and suppose the sequence 0 We suppose that I ⊆ J ob and derive a contradiction.Fix a ∈ I \ J ob .In particular, and contains x 0 .By Urysohn's lemma we may pick a function u ∈ C 0 (G (0) ) such that 0 u 1, u(x 0 ) = 1, and u vanishes outside V .Set z := ua * a ∈ I \ J ob and observe that E(z) = uE(a * a) = uf , and We claim that there exists h ∈ C 0 (G (0) ) satisfying 0 h 1, h(x 1 ) = 1 and For each i such that B i ∩ I(G) x 0 = ∅, we can apply Lemma 4.10 to obtain a function and for each i such that For each i such that B i ∩ I(G) x 0 = ∅, Lemma 4.10 for B i at x 1 yields a function Altogether we have constructed functions h 1 , . . ., h k that all satisfy (4.8).Set h := ) and note that 0 h 1 and h(x 1 ) = 1.It remains to verify (4.4) and (4.5); we do this by direct computation.Using (4.3) and the fact that u(x 0 ) = 1, we see that By first using the choice of g and then the choice of x 1 from (4.7), we find Remembering that h(x 1 ) = 1, we obtain This means that E(z) − ε < hE(z)h so (4.4) follows.For (4.5), we use the decomposition (4.6) and then (4.8) to see that and this proves (4.5).
The lemma below uses the full groupoid C * -algebra C * (G).We refer the reader to [Wil19] for a discussion of this C * -algebra that does not assume second-countability.Lemma 4.13.Let G be a locally compact Hausdorff étale groupoid whose full and reduced C * -algebras coincide.Let J ob be the obstruction ideal in C * r (G).There is a *representation ε of the full groupoid C * -algebra C * (G) such that ker(ε) is purely nondynamical and such that supp(J ob ) ⊆ supp(ker(ε)).
To see that supp(ker(ε)) contains supp(J ob ), by Lemma 2.1 it suffices to show that Corollary 4.14.Let G be a locally compact Hausdorff étale groupoid that is jointly effective where it is effective.Suppose the sequence 0 → J ob → C * r (G) → C * r (G eff ) → 0 is exact and that the full and reduced groupoid C * -algebras of G coincide.Then there is a purely non-dynamical ideal whose support is equal to that of J ob , and J ob is the minimal dynamical ideal that contains all purely non-dynamical ideals of C * r (G).Proof.Lemma 4.13 gives a purely non-dynamical ideal I such that supp(J ob ) ⊆ supp(I).Theorem 4.12 shows that J ob contains all purely non-dynamical ideals in C * r (G), and in particular contains I. Hence supp(I) ⊆ supp(J ob ), and we obtain equality.Now suppose that I U is a dynamical ideal that contains every purely non-dynamical ideal.Then in particular, To finish the section, we observe that our results can be used to recover [BCFS, Proposition 5.5(2)], without the assumption that G is second-countable.This is not new.For example, it can be recovered from a special case of [KM19,Theorem 7.29].We include it here only to illustrate how our results relate to effective groupoids.Proposition 4.15.Let G be a locally compact Hausdorff étale groupoid.
(1) If G is effective, then every nontrivial ideal of C * r (G) contains a nonzero element of C 0 (G (0) ).
(2) If every nontrivial ideal of the full C * -algebra C * (G) contains a nonzero element of C 0 (G (0) ), then the full and reduced C * -algebras of G coincide and G is effective. Proof.
(1) Fix an ideal I of C * r (G) that contains no nonzero element of C 0 (G (0) ); we must show that I = {0}.Since G is effective, it is jointly effective where it is effective, and J ob is trivial.The sequence 0 → J ob → C * r (G) → C * r (G eff ) → 0 is then trivially exact, and Theorem 4.12 implies that I ⊆ J ob = {0}.
(2) We prove the contrapositive.First suppose that the full and reduced C * -algebras of G do not coincide.Then the kernel of the regular representation λ : C * (G) → C * r (G) is a nonzero purely non-dynamical ideal.Now suppose that the full and reduced C * -algebras of G coincide but that G is not effective.Then J ob is nontrivial, and Lemma 4.13 implies that there is a purely non-dynamical ideal I of C * r (G) whose support contains that of J ob , and in particular is nonzero.
Remark 4.16.A mainstay of the theory of étale groupoid C * -algebras is the diagonal uniqueness theorem, dating back to [Re80]: for amenable effective étale groupoids, any * -homomorphism that is injective on the diagonal is injective (See Proposition 4.15).If G is a groupoid that does not satisfy the conclusion of this theorem, then there is a * -homomorphism φ of C * r (G) whose kernel is purely non-dynamical.So if G is also innerexact Hausdorff étale groupoid whose full and reduced C * -algebras coincide, then the kernel of φ is contained in the obstruction ideal.This justifies the terminology obstruction ideal : the obstruction ideal measures how far away a groupoid is from satisfying a diagonal uniqueness theorem.
For example, if G is the groupoid of a higher-rank graph in the sense of [KP00], then the obstruction ideal is zero if and only if the higher-rank graph is aperiodic (so its C * -algebra satisfies the Cuntz-Krieger uniqueness theorem) [RS07, Proposition 3.6].

Examples
5.1.Groupoids from local homeomorphisms.First we consider the groupoid constructed from a local homeomorphisms T on a locally compact Hausdorff space X.The associated semi-direct product groupoid, usually called the Deaconu-Renault groupoid, is where the product of (x, p, y) and (y ′ , q, z) is defined precisely if y = y ′ in which case (x, p, y)(y, q, z) = (x, p + q, z) while inversion is (x, p, y) −1 = (y, −p, x).The unit space is naturally identified with X and the range and source maps are then r(x, p, y) = x and s(x, p, y) = y.
We first verify that this groupoid is jointly effective where it is effective.For open subsets U and V of X, the sets of the form For the rank-one Deaconu-Renault groupoids, we can describe explicitly the points that are not effective .For p ∈ N + , let P p = {x ∈ X : T p pointwise fixes a neighbourhood of x} (5.1) and let P = ∞ p=1 P p .Then P is open and invariant in X and the restricted system (P, T ) is reversible.(5.3)Note that B ⊆ I • (G) \ X.In particular, I • (G) x ′ contains (x ′ , p, x ′ ), so x ′ is not effective, and by invariance x is not effective.
For the other inclusion, suppose x is not effective.Then (x, p, x) ∈ I • (G) x for some p ∈ N + .Fix an open bisection B in I • (G) \ X containing (x, p, x).We may assume that T p x = x, so by shrinking B we may assume that B ⊆ Z(U, p, 0, U) for some open subset U of X.Then T p y = y for every y ∈ s(B) since B ⊆ I(G).Therefore, x ∈ V .
Next we show that any Deaconu-Renault groupoid G T is jointly effective where it is effective.The result actually only depends on the nontrivial isotropy being infinite cyclic, so we record this more general result here.
Lemma 5.2.Any Hausdorff étale groupoid G whose nontrivial isotropy is infinite cyclic is jointly effective where it is effective.
Proof.Let x ∈ G (0) be a point with nontrivial isotropy and suppose B 1 , . . ., B N are open bisections in G such that each B i contains an element γ i ∈ Iso(G) x \ G (0) .Since the isotropy group at x is infinite cyclic there are minimal integers p 1 , . . ., p N such that γ p i i = γ p j j for all i, j = 1, . . ., N. Put γ := γ p i i .Then B := N is an open bisection containing γ.
Assume now that x is effective.So whenever U ⊆ G (0) is an open neighbourhood of x, there is a point y ∈ U such that r(By) = y.Applying this to a neighbourhood basis of x, we can find a sequence (y n ) n in G (0) such that y n → x and r(By n ) = y n for all n.We show that G is jointly effective at x.It suffices to show that for large n, we have r(B i y n ) = y n for all i = 1, . . ., N.
Since B p 1 1 contains γ, we have x ∈ s(B p 1 1 ) so y n ∈ s(B p 1 1 ) for large n, and since B ⊆ B p 1 1 we see that r(B p 1 1 y n ) = r(By n ).If r(B 1 y n ) = y n , then y n = r(B 1 y n ) = r(B p 1 1 y n ) = r(By n ), which contradicts our choice of y n .So for large n we have r(B 1 y n ) = y n as required.Hence G is jointly effective at x.
As an immediate corollary we see that the groupoids built from a local homeomorphism T on a locally compact Hausdorff space X, called rank-one Deaconu-Renault groupoids, are covered by the above result.
Corollary 5.3.Any rank-one Deaconu-Renault groupoid is jointly effective where it is effective.5.2.Partial actions.Our notion of being jointly effective for groupoids is directly inspired by Ara and Lolk's notion of relative strong topological freeness for partial actions [AL18, Section 7].A partial action θ : Γ X of a countable discrete group Γ on a locally compact Hausdorff space X is topologically free at x ∈ X if whenever θ g (x) = x for some 1 = g ∈ Γ, for any open neighbourhood U of x, there exists y ∈ U such that θ g (y) = y.We say θ is strongly topologically free at x if for any finite collection 1 = g 1 , • • • , g k ∈ Γ such that θ g i (x) = x and any neighbourhood U around x, there exists y ∈ U such that θ g i (y) = y for all i = 1, • • • , k.Finally, θ is relatively strong topologically free if it is strongly topologically free at all points at which it is topologically free.
Following [Ab04, Section 2], a partial action θ : Γ X has an associated groupoid whose unit space G (0) θ is naturally identified with X. Elements (x, g, y) and (y ′ , g ′ , z) in G θ are composable if and only if y = y ′ in which case (x, g, y)(y ′ , g ′ , z) = (x, gg ′ , z).Inversion is given by (x, g, y) −1 = (y, g −1 , x).The source and range maps s, r : G θ → X are s(x, g, y) = y and r(x, g, y) = x.The groupoid G θ carries a locally compact and Hausdorff étale topology.
Lemma 5.4.Let θ : Γ X be a partial action of a countable discrete group Γ on a locally compact Hausdorff space X.Then θ is topologically free at x ∈ X if and only if G θ is effective at x.Moreover, θ is strongly topologically free at x if and only if G θ is jointly effective at x.In particular, θ is relatively strongly topologically free if and only if G θ is jointly effective where it is effective.
Proof.Suppose that θ is not strongly topologically free at x.There exist g 1 , . . ., g k ∈ Γ \ {e} that all fix x, and a neighbourhood U of x such that for every y ∈ U there exists i such that θ g i (y) = y.For each i, define B i := {(θ g i (y), g i , y) : y ∈ U}.
Then each B i is a bisection containing (x, g i , x), and there is no y ∈ i s(B i ) = U such that r(B i y) = y for all i.So G θ is not jointly effective at x. Taking k = 1 shows that if θ is not topologically free at x then G θ is not effective at x. Now suppose that G θ is not jointly effective at x. Fix elements γ 1 , . . ., γ k ∈ I(G θ ) x \{x} and open bisections B i containing γ i such that for each y ∈ i s(B i ) there exists i such that r(B i y) = y.By definition of G θ , each γ i = (x, g i , x) for some g i ∈ Γ \ {e}.By definition of the topology on G θ , for each i there is an open neighborhood U i of x such that {(θ g i (y), g i , y) : y ∈ U i } ⊆ B i .Now U = i U i is a neighbourhood of x and for each y ∈ U there exists i such that r(B i y) = y.That is, θ g i (y) = y.So θ is not strongly topologically free at x. Again, taking k = 1 throughout shows that if G θ is not effective at x then θ is not topologically free at x.
The final statement follows by definition.
an action by d local homeomorphisms, then we let G T denote the Deaconu-Renault groupoid of T as in, for example, [SW16, Section 3].An ideal I of C * (G T ) is gauge-invariant if the canonical gauge action γ of T d on C * r (G T ) satisfies γ z (I) ⊆ I for all z ∈ T d .Proposition 3.9.Let X be a locally compact Hausdorff space and suppose T : N d X is an action on X by d commuting local homeomorphisms.The map that carries an open invariant subset U of X to the ideal I U generated by C 0 (U) is a lattice isomorphism from the lattice of open invariant subsets of X to the lattice of gauge-invariant ideals of C * (G T ).
fix x ∈ U. Since G (0) eff is invariant, V := r(GU) is open and invariant with U ⊆ V ⊆ G (0) be the function given by f (η) = f (s(η)) for all η ∈ B. By extending by zero, both functions can be regarded as elements of C c (G). Direct calculation on basis elements (see the proof of [BCFS, Proposition 5.5(2)]) shows that f − f ∈ ker(ε).So x ∈ supp(ker(ε)).
comprise a basis for a locally compact Hausdorff étale topology on G T .The groupoid G T is amenable, and hence inner-exact [SW16, Section 3].

Lemma 5. 1 .
For a local homeomorphism T on a locally compact Hausdorff space X, we have X \ X eff = {x ∈ X : orb T (x) ∩ P = ∅}.(5.2)Proof.Let V := {x ∈ X : orb T (x) ∩ P = ∅} and let G = G T be the Deaconu-Renault groupoid of T .It is straightforward to verify that V is open and invariant in X.We verify (5.2) one inclusion at a time.Let x ∈ V and choose l ∈ N such that x ′ := T l (x) ∈ P p for some p ∈ N + .Pick an open set U ⊆ X all of whose points are p-periodic and consider the open bisection given by B = {(y, p, y) ∈ G : y ∈ U}.