Cordeiro Luiz
email: luizgc6@gmail.com

Beuter Viviane

Gustavo Luiz

Cordeiro

Viviane Beuter
email: vivibeuter@gmail.com

The dynamics of partial inverse semigroup actions

Keywords: Mathematics Subject Classication. Primary 20M30; Secondary 16S99, 22A22 Partial action of inverse semigroups, groupoid of germs, Steinberg algebras, crossed product, skew inverse semigroup ring, topologically free, topologically principal, eective, continuous orbit equivalence, E-unitary

come

Introduction

Partial actions of groups on C*-algebras, initially dened for the group of integers in [START_REF] Exel | Circle actions on C * -algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence[END_REF] (and for general discrete groups in [START_REF] Mcclanahan | K-theory for partial crossed products by discrete groups[END_REF]), are a powerful tool in the study of many C*-algebras associated to dynamical systems.

In [START_REF] Dokuchaev | Associativity of crossed products by partial actions, enveloping actions and partial representations[END_REF], Dokuchaev and Exel introduced the analogous notion of partial group actions in a purely algebraic context, and although the theory is not at present as well-developed as its C*-algebraic counterpart, it has attracted interest or researchers in the area, since some important classes of algebra, such as graph and ultragraph Leavitt path algebras, have been shown to be crossed products (see [START_REF]Leavitt path algebras as partial skew group rings[END_REF][START_REF]Simplicity and chain conditions for ultragraph Leavitt path algebras via partial skew group ring theory[END_REF]).

In fact, in [START_REF]Partial actions of groups and actions of inverse semigroups[END_REF]Theorem 4.2] it is proven that partial group actions correspond to actions of certain universal inverse semigroups, which were already considered in [START_REF] Sieben | C * -crossed products by partial actions and actions of inverse semigroups[END_REF] and can be used, for example, to describe groupoid C*-algebras as crossed products by inverse semigroups (see [START_REF] Alan | Groupoids, inverse semigroups, and their operator algebras[END_REF]Theorem 3.3.1]). Although these approaches are similar in some respects, each of them has its advantages and drawbacks for example, actions of inverse semigroups respect the operation completely, whereas groups have, overall, a better algebraic structure.

Groupoids are also being extensively used in order to classify and study similar classes of algebras (see [START_REF] Orlo | A groupoid generalisation of Leavitt path algebras[END_REF] for example), and one can relate these two approaches in the following manner: From a partial group action on a topological space we associate a transformation groupoid, or from an inverse semigroup action on a space we associate a groupoid of germs (see [START_REF] Abadie | On partial actions and groupoids[END_REF] and [START_REF]Inverse semigroups and combinatorial C * -algebras[END_REF], respectively). It turns out that both in the purely algebraic and the C*-algebraic settings, the algebras of such groupoids coincide with the algebras induced from the group or semigroup actions (see [START_REF]The interplay between Steinberg algebras and skew rings[END_REF][START_REF] Demeneghi | The ideal structure of Steinberg algebras[END_REF]). In fact, under appropriate assumptions, the relationships between the representation theory of groupoids and inverse semigroups have also been made categorical, see for example [START_REF] Buss | Inverse semigroup actions as groupoid actions[END_REF][START_REF] Lawson | Pseudogroups and their étale groupoids[END_REF][START_REF] Bice | General non-commutative locally compact locally Hausdor Stone duality[END_REF].

In this article we will be concerned with partial actions of inverse semigroups, dened in [START_REF] Buss | Inverse semigroup expansions and their actions on C * -algebras[END_REF], which are a common generalization of both partial actions of groups and actions of inverse semigroups. In particular, we generalize the constructions of groupoids of germs for topological partial actions, and of crossed products for algebraic partial actions.

Therefore we have a common ground for the study of both partial group actions and inverse semigroup actions.

The rst problem we tackle is to describe the Steinberg algebra of the groupoid of germs of a topological partial inverse semigroup action as a crossed product algebra. This generalizes results of [START_REF]The interplay between Steinberg algebras and skew rings[END_REF][START_REF] Demeneghi | The ideal structure of Steinberg algebras[END_REF], where such isomorphisms were obtained under (strictly) stronger assumptions. In the converse direction, by starting with an appropriate crossed product, we manage to construct a groupoid of germs which realizes the isomorphism above.

Orbit equivalence and full groups for Cantor systems were initially studied by Giordano, Putnam and Skau in [START_REF] Giordano | Topological orbit equivalence and C * -crossed products[END_REF][START_REF]Full groups of Cantor minimal systems[END_REF], and has enjoyed recent developments in [START_REF]Continuous orbit equivalence rigidity[END_REF][START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF][START_REF] Brownlowe | Graph algebras and orbit equivalence[END_REF]. The notion of continuous orbit equivalence can be immediately extended to partial inverse semigroup actions. We introduce and study a natural notion of topological principality for partial inverse semigroup actions, which corresponds to topological principality of the groupoid of germs. In the Hausdor setting, we prove that two ample, topologically principal partial inverse semigroup actions are continuously orbit equivalent if and only if the corresponding groupoids of germs are isomorphic, thus generalizing the analogous part of [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF]Theorem 2.7]. It is important to note that the semigroups considered do not need to be isomorphic, since continuous orbit equivalence deals mostly with the dynamics of the unit space inherited from the partial action.

We nish this article by connecting the notions of continuous orbit equivalence of (partial) semigroup actions, continuous orbit equivalence of graphs, and isomorphism of Leavitt path algebras.

Given an inverse semigroup S, we denote by E(S) = e ∈ S : e 2 = e the set of idempotents of S. E(S) is a commutative sub-inverse semigroup of S, and it is a semilattice (Denition 3.21) under the order e ≤ f ⇐⇒ e = ef . This order is extended to all of S by setting s ≤ t ⇐⇒ s = ts * s. This order is preserved under products and inverses of S. Homomorphisms of inverse semigroups preserve their orders.

Partial actions of inverse semigroups. Denition 2.2 ([8, Denition 2.11, Proposition 3.1]). A partial homomorphism between inverse semigroups S and T is a map ϕ : S → T such that for all s and t in S, one has that (i) ϕ(s * ) = ϕ(s) * ;

(ii) ϕ(s)ϕ(t) ≤ ϕ(st);

(iii) ϕ(s) ≤ ϕ(t) whenever s ≤ t.

Note that homomorphisms of inverse semigroups are also partial homomorphisms.

In the most general context ([8, Denition 3.3]), a partial action of a semigroup S on a set X is simply a partial homomorphism S → I(X). However, when X has some extra structure (topological and/or algebraic) we will be interested in partial actions that preserve this structure. Denition 2.3. A topological partial action of an inverse semigroup S on a topological space X is a tuple θ = ({X s } s∈S , {θ s } s∈S ) such that:

(i) For all s ∈ S, X s is an open subset of X and θ s : X s * → X s is a homeomorphism; (ii) The map s → θ s is a partial homomorphism of inverse semigroups; (iii) X = e∈E(S) X e .

If s → θ s is a homomorphism of inverse semigroups, we call θ a global action, or simply an action if no confusion arises. Condition (iii) above is usually called non-degeneracy, and is sometimes not required. If (i) and (ii) are satised by a tuple θ as above, then X s * ⊆ X s * s for all s ∈ S, and thus one can always substitute X by e∈E(S) X e (which in fact coincides with s∈S X s ) and obtain a non-degenerate partial action. In fact, θ is a global action if and only if X s * = X s * s for all s ∈ S. Similar comments hold for partial actions of groups on algebras, which we now dene. For the remainder of this section, we x a commutative unital ring R, and will consider algebras over R. Remark 2.4. Every ring has a canonical Z-algebra structure, or alternatively, when restricted to commutative rings, every commutative ring R has a canonical R-algebra structure. Thus the denitions we adopt for algebras restrict to partial actions and crossed products of rings. Denition 2.5. An algebraic partial action of an inverse semigroup S on an associative R-algebra A is a tuple α = ({A s } s∈S , {α s } s∈S ) such that:

(i) For all s ∈ S, A s is an ideal of A and α s : A s * → A s is an R-isomorphism;

(ii) α : S → I(A), s → α s is a partial homomorphism of inverse semigroups;

(iii) X = span R e∈E(S) A e .

If s → α s is a homomorphism of inverse semigroups, we call α a global action or simply an action.

Crossed products. Let R be a commutative unital ring, and let α = ({A s } s∈S , {α s } s∈S ) be a partial action of an inverse semigroup S on an associative R-algebra A. Consider L = L (α) the R-module of all nite sums of the form nite s∈S a s δ s , where a s ∈ A s and δ s is a formal symbol, with a multiplication dened as the bilinear extension of the rule (a s δ s )(b t δ t ) = α s (α s * (a s )b t )δ st .

Then L is an R-algebra which is possibly not associative (see [START_REF] Dokuchaev | Associativity of crossed products by partial actions, enveloping actions and partial representations[END_REF]Example 3.5]). A proof similar to that of [START_REF] Dokuchaev | Associativity of crossed products by partial actions, enveloping actions and partial representations[END_REF]Corollary 3.2] shows that if A s is idempotent or non-degenerate for each s ∈ S, then L is associative. Denition 2.6. Let α = ({A s } s∈S , {α s } s∈S ) be an algebraic partial action of an inverse semigroup S on an R-algebra A end let N = N (α) be the additive subgroup of L generated by all elements of the form aδ r -aδ s , where r ≤ s and a ∈ A r .

Then N is an ideal of the R-algebra L . We dene the crossed product, which we denote by A α S (or simply A S) as the quotient algebra A α S := L /N

The class of an element x ∈ L in A α S will be denoted by x.

Remark 2.7.

(1) As a ring, A S depends only on the ring structure of A and the maps α. So distinct algebra structures over A will induce distinct algebra structures over the same ring A S (as long as the partial action preserves these distinct algebra structures).

(2) Crossed products are sometimes called skew inverse semigroup algebras or rings, or partial crossed products (see [START_REF] Beuter | Simplicity of skew inverse semigroup rings with applications to steinberg algebras and topological dynamics[END_REF][START_REF] Boava | Partial crossed product description of the C * -algebras associated with integral domains[END_REF][START_REF] Dokuchaev | Associativity of crossed products by partial actions, enveloping actions and partial representations[END_REF]). Since these are simply particular cases of the construction above, we adopt the simplest nomenclature for the most general case.

The diagonal of the crossed product A S is the additive abelian subgroup generated by elements of the form aδ e , where e ∈ E(S) and a ∈ A e , and the diagonal is a subalgebra of A S.

Recall that a ring B is left s-unital if for all nite subsets F ⊆ B, there exists u ∈ B such that x = ux for all x ∈ F . Proposition 2.8. Suppose that α is a partial action of S on an algebra A, and that for all e ∈ E(S), A e is a left s-unital ring. Then A is isomorphic to the diagonal algebra of A S.

Proof. Any element of A is a sum of elements of e∈E(S) A e , so the same argument of [START_REF] Tominaga | On s-unital rings[END_REF]Theorem 1] proves that A is a left s-unital ring. The proof of [START_REF]Socity and other dynamical aspects of groupoids and inverse semigroups[END_REF]Proposition 4.3.11] (see also [START_REF] Beuter | Simplicity of skew inverse semigroup rings with applications to steinberg algebras and topological dynamics[END_REF]Proposition 3.1]) can be easily adapted to obtain an isomorphism between A and the diagonal subalgebra of A S.

Étale Groupoids. A groupoid is a small category G with invertible arrows. We identify G with the underlying set of arrows, so that objects of G correspond to unit arrows, and the space of all units is denoted by G (0) . The source of an element a ∈ G is dened as s(a) = a -1 a and the range of a is r(a) = aa -1 . A pair (a, b) ∈ G × G is composable (i.e., the product ab is dened) if and only if s(a) = r(b), and the set of all composable pairs is denoted by G (2) .

A topological groupoid is a groupoid G endowed with a topology which makes the multiplication map G (2) (a, b) → ab ∈ G and the inverse map G a → a -1 ∈ G continuous, where we endow G (2) with the topology induced from the product topology of G × G. Denition 2.9 ([36, 49]). An étale groupoid is a topological groupoid G such that the source map s :

G → G (0) is a local homeomorphism.
Alternatively, a topological groupoid G is étale precisely when G (0) is open and the product of any two open subsets of G is open (see [START_REF] Resende | étale groupoids and their quantales[END_REF]Theorem 5.18]), where the product of A, B ⊆ G is dened as (2) .

AB = ab : (a, b) ∈ (A × B) ∩ G
An open bisection of an étale groupoid is an open subset U ⊆ G such that the source and range maps are injective on U , and hence homeomorphisms onto their images. The set of all open bisections of an étale groupoid forms a basis for its topology and it is an inverse semigroup under the product of sets. We denote this semigroup by G op . Denition 2.10. An étale groupoid is ample if G (0) is Hausdor and admits a basis of compact-open subsets.

Suppose that G is an ample groupoid. Then G admits a basis of compact-open bisections, since G (0) does and s : G → G (0) is a local homeomorphism. Since G (0) is Hausdor then G (2) is closed in G × G, then the product of two compact subsets A, B of G is compact, as AB is the image of the compact (A × B) ∩ G (2) under the continuous product map (alternatively, see [START_REF] Lawson | Pseudogroups and their étale groupoids[END_REF]Lemma 3.13]). We denote by G a the sub-inverse semigroup of G op consisting of compact-open bisections and call G a the ample semigroup of G.

Example 2.11. Let G be an étale groupoid. The canonical action of G op on G (0) is dened as

τ = τ G = {r(U )} U ∈G op , {τ U } U ∈G op , with τ U : s(U ) → r(U ) the homeomorphism τ = r • s | -1
U . This is the homeomor- phism which takes the source of each arrow of U to its range.

Steinberg algebras of ample groupoids. Throughout this section, we x a commutative unital ring R.

Given an ample Hausdor groupoid G, we denote by R G the R-module of R-valued functions on G. Given A ⊆ G, dene 1 A as the characteristic function of A (with values in R).

Denition 2.12. Given an ample groupoid

G, A R (G) is the R-submodule of R G generated by the charac- teristic functions of compact-open bisections of G. The support of f ∈ R G is dened as supp f = {a ∈ G : f (a) = 0}. If G is Hausdor, then A R (G) coincides with the R-module of locally constant compactly supported R-valued functions on G [12, Lemma 3.3].
In the general (non-Hausdor ) case, for every f ∈ A R (G) and every x ∈ G (0) , (supp f ) ∩ s -1 (x) and (supp f ) ∩ r -1 (x) are nite, and so we can dene their convolution product

(f * g)(a) = xy=a f (x)g(y) = x∈r -1 (r(a)) f (x)g(x -1 a) = y∈s -1 (s(a)) f (ay -1 )g(y). This product makes A R (G) an associative R-algebra, called the Steinberg algebra of G (with coecients in R). The map G a → A R (G), U → 1 U , is a representation of G a as a Boolean semigroup (see [36]), that is, it satises (i) 1 U * 1 V = 1 U V ; and (ii) 1 U ∪V = 1 U + 1 V if U ∩ V = ∅ and U ∪ V ∈ G a .
In fact, A R (G) is universal for such representations. The proof for a general commutative ring with unit R follows the same arguments as in [START_REF] Orlo | A groupoid generalisation of Leavitt path algebras[END_REF]Theorem 3.10], and we state it here explicitly: Theorem 2.13 (Universal property of Steinberg algebras, [START_REF]Socity and other dynamical aspects of groupoids and inverse semigroups[END_REF]Theorem 4.4.8]). Let R be a commutative unital ring and G an ample Hausdor groupoid. Then A R (G) is universal for Boolean representations of G a , i.e., if B is an R-algebra and π :

G a → B is a function satisfying (i) π(AB) = π(A)π(B) for all A, B ∈ G a ; and (ii) π(A) = π(A \ B) + π(B) whenever A, B ∈ G a and B ⊆ A, then there exists a unique R-algebra homomorphism Φ : A R (G) → B such that Φ(1 U ) = π(U ) for all U ∈ G a .
Recall that a topological space X is zero-dimensional (or has small inductive dimension 0) if it admits a basis of clopen subsets of X. A locally compact Hausdor space is zero-dimensional if and only if it is totally disconnected. Moreover, an étale groupoid G is ample if and only if G (0) is locally compact Hausdor and zero-dimensional.

Example 2.14. Every locally compact Hausdor and zero-dimensional space X is an ample groupoid with X (0) = X (that is, the product is only dened as xx = x for all x ∈ X). The Steinberg algebra A R (X) coincides with the R-algebra of locally constant compactly supported R-valued functions on X with pointwise operations.

In general, we identify

A R (G (0) ) with the sub-R-algebra D R (G) = f ∈ A R (G) : supp f ⊆ G (0) of A R (G), called the diagonal subalgebra of A R (G). More precisely, the map D R (G) f → f | G (0) ∈ A R (G (0)
) is an R-algebra-isomorphism, and its inverse extends every f ∈ A R (G (0) ) as 0 on G \ G (0) .

Groupoids of germs

Groupoids of germs were already considered by Paterson in [START_REF] Alan | Groupoids, inverse semigroups, and their operator algebras[END_REF] for localizations of inverse semigroups, and for natural actions of pseudogroups by Renault in [START_REF]Cartan subalgebras in C * -algebras[END_REF]. In [START_REF]Inverse semigroups and combinatorial C * -algebras[END_REF], Exel dened groupoids of germs for arbitrary actions of inverse semigroups on topological spaces in a similar, albeit more general, manner than both previous denitions of groupoids of germs. Moreover, partial actions of groups also introduced by Exel in [START_REF] Exel | Circle actions on C * -algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence[END_REF] have many application in the theory of C*-dynamics, see for example [START_REF] Boava | Partial crossed product description of the C * -algebras associated with integral domains[END_REF][START_REF] Toke | Partial actions and KMS states on relative graph C * -algebras[END_REF][START_REF] Exel | Cuntz-Krieger algebras for innite matrices[END_REF][START_REF] Giordano | Bratteli-Vershik models for partial actions of Z[END_REF][START_REF] Gonçalves | C*-algebras associated to stationary ordered Bratteli diagrams[END_REF][START_REF]Ultragraphs and shift spaces over innite alphabets[END_REF].

Partial group actions also induce transformation groupoids similarly to the classical (global) case, see [START_REF] Abadie | On partial actions and groupoids[END_REF].

Our objective in this section is to construct a groupoid of germs associated to any partial action of an inverse semigroup in a way that generalizes both groupoids of germs of inverse semigroup actions, and transformation groupoids of partial group actions.

Let θ = ({X s } s∈S , {θ s } s∈S ) be a partial action of an inverse semigroup S on a topological space X. We denote by S * X the subset of S × X given by S * X := {(s, x) ∈ S × X : x ∈ X s * } .

Recall that a semigroupoid is a structure satisfying the same axioms as a category 1 , except possibly the existence of identities at objects (see [START_REF] Tilson | Categories as algebra: an essential ingredient in the theory of monoids[END_REF]Appendix B]). Quotients of semigroupoids are dened, up to obvious modications, in the same manner as quotients of categories (see [START_REF] Mac | Categories for the working mathematician[END_REF]Section II.8]).

We make S * X a semigroupoid with object space (S * X) (0) = X by setting the source and range maps as s(s, x) = x, r(s, x) = θ s (x)

and the product (s, x)(t, y) = (st, y) whenever s(s, x) = r(t, y). Moreover, S * X is an inverse semigroupoid, in the sense that for every p = (s, x) ∈ S * X, p * = (s * , θ s (x)) is the unique element of S * X satisfying pp * p = p and p * pp * = p * . We dene the germ relation ∼ on S * X: for every (s, x) and (t, y) in S * X,

(s, x) ∼ (t, y) ⇐⇒ x = y and there exists u ∈ S such that u ≤ s, t and x ∈ X u * .

Alternatively,

(s, x) ∼ (t, y) ⇐⇒ x = y and there exists e ∈ E(S) such that x ∈ X e and se = te. Proof. The relation ∼ is clearly reexive and symmetric. As for transitivity, if (s, x) ∼ (t, y) and (t, y) ∼ (r, z), then x = y = z, so there exist u ≤ s, t and v ≤ t, r such that x ∈ X u * ∩ X v * . It follows that

uv * v ≤ tv * v = v ≤ r, and of course uv * v ≤ u ≤ s. Moreover, x ∈ X u * ∩ X v * v ⊆ X (uv * v) * . Therefore (s, x) ∼ (r, z) by (3.1).
To prove that ∼ is a congruence, rst note that the source and range maps of S * X are invariant on ∼-equivalence classes. We need to prove that ∼ is invariant under taking products, so suppose that (s i , x i ) ∼ (t i , y i ) (i = 1, 2) and s(s 1 , x 1 ) = r(s 2 , x 2 ). Then for i = 1, 2 we have x i = y i , and there exists

u i ≤ s i , t i such that x i ∈ X u * i . Thus u 1 u 2 ≤ s 1 s 2 , t 1 t 2 and x 2 ∈ X u * 2 ∩ θ -1 u2 (X u * 1 ∩ X u2 ) ⊆ X (u1u2) * , because θ u2 (x 2 ) = θ s2 (x 2 ) = x 1 . This proves that (s 1 , x 1 )(s 2 , x 2 ) = (s 1 s 2 , x 2 ) ∼ (t 1 t 2 , y 2 ) = (t 1 , y 1 )(t 2 , y 2 ).
To conclude that S X is a groupoid, simply note that for all (s, x) Note that, by construction, the object (unit) space of S X is X, which we identify with the set of unit arrows of S X as usual: the source s[s, x] = x of an arrow [s, x] ∈ S X corresponds to the arrow

∈ S * X, if (s, x)(t,
[s, x] -1 [s, x] = [s * , θ s (x)][s, x] = [s * s, x]
and similarly the range r[s, x] = θ s (x) corresponds to the arrow [ss * , θ s (x)]. In other words, we identify (S X) (0) and X via the map

(3.5) X → (S X) (0) , x → [e, x]
, where e ∈ E(S) is chosen so that x ∈ X e which is well-dened since we only consider non-degenerate partial actions.

We will now endow S X with an appropriate topology. Given s ∈ S and U ⊆ X s * , dene the subset

[s, U ] of S X [s, U ] = {[s, x] : x ∈ U } .
Using the denition of germs, it readily follows that Let m : (S X) (2) → S X be the product map. Given [u, U ] ∈ B germ , let us prove that (2) : s, t ∈ S and st ≤ u Indeed, the inclusion ⊇ in (3.8) is immediate from the denition of the product. For the converse inclusion, assume ([s, y],

(3.6) [s, U ] ∩ [t, V ] = {[z, U ∩ V ∩ X z * ] : z ∈ S,
(3.8) m -1 [u, U ] = ([s, X s * ] × [t, U ∩ X t * ]) ∩ (S X)
[q, x]) ∈ m -1 [u, U ]. This means that [sq, x] ∈ [u, U ], so x ∈ U and there exists v ≤ sq, u such that x ∈ X v * . Set t = qv * v, so that x ∈ X t * , [q, x] = [t,
x], and st = sqv * v = v ≤ u, and therefore ([s, y], [q, x]) = ([s, y], [t, x]) belongs to the right-hand side of (3.8). This proves that the product map is continuous.

Similarly, the denition of the inverse in S X implies that

[u, U ] -1 = [u * , θ u (U )] ∈ B germ ,
so the inversion map is also continuous. The proof above shows that a basic open set of (S X) (0) is of the form [e, U ], where e ∈ E(S) and U ⊆ X e . Under the identication of (S X) (0) with X as in Equation (3.5), [s, U ] corresponds to U . Therefore this identication is a homeomorphism. Notice that if s ∈ S and U ⊆ X s * is an open set then [s, U ] is compact if and only if U is compact. Moreover, if B is a basis for the topology of X, then a basis for S X consists of those sets of the form [s, U ] with U ∈ B. Hence, if X is zero-dimensional then the collection of sets of the form [s, U ] with U compact-open subset of X is a basis for S X.

Corollary 3.10. If X is a locally compact Hausdor and zero-dimensional space then S X is an ample groupoid.

Let us prove a universal property for the groupoid of germs. Recall from Example 2.11 the denition of the canonical action of an étale groupoid. Theorem 3.11 (Universal property of groupoids of germs). Let θ be a topological partial action of an inverse semigroup S on a topological space X. Suppose that G is an étale groupoid, σ : S → G op is a partial homomorphism, and φ : X → G (0) is a continuous function satisfying (i) φ(X s ) ⊆ r(σ(s)) for all s ∈ S; and (ii) τ σ(s) (φ(x)) = φ(θ s (x)) for all s ∈ S and x ∈ X s * , where τ denotes the canonical action of G op on G (0) . Then there exists a unique continuous groupoid homomorphism Ψ : S X → G satisfying (3.12)

Ψ[s, x] ∈ σ(s) and s(Ψ[s, x]) = φ(x), whenever s ∈ S and x ∈ X s * . Proof. Equation (3.12) simply means that Ψ[s, x] = s | -1 σ(s) (φ(x)) for all s ∈ S and x ∈ X s * , so uniqueness is immediate. Dene Φ : S * X → G by Φ(s, x) = s | -1 σ(s) (φ(φ(x))) for all (s, x) ∈ S * X, that is, Φ(s, x)
is the arrow in σ(s) with source φ(x). Let us prove that it is a semigroupoid homomorphism.

Suppose that the product (s, x)(t, y) is dened in S * X. This means that x = θ t (y). Applying φ and using property ((ii)) yields

s(Φ(s, x)) = φ(x) = φ(θ t (y)) = τ σ(t) (φ(y)),
and the last term above is simply the range of the arrow in σ(t) whose source is φ(y), that is,

s(Φ(s, x)) = r(Φ(t, y)).
Therefore, the product Φ(s, x)Φ(t, y) is dened. It belongs to σ(s)σ(t)σ(st), since σ is a partial homomorphism, and its source is s(Φ(t, y)) = φ(y). Therefore,

Φ(s, x)Φ(t, y) = Φ(st, y) = Φ((s, x)(t, y)),
which proves that Φ is a semigroupoid homomorphism. Let us prove that Φ is invariant by the germ relation ∼ as in (3.1): Suppose (s, x) ∼ (t, y). Then x = y and there exists v ≤ s, t such that x ∈ X v * . Then Φ(v, x) is an arrow which in σ(v) ⊆ σ(s), σ(t), as σ is a partial homomorphism, and whose source is φ(x), thus

Φ(s, x) = Φ(v, x) = Φ(t, x) = Φ(t, y).
Therefore Φ factors though a groupoid homomorphism Ψ : S X → G satisfying (3.12).

It remains only to prove that

Ψ is continuous. Suppose that V ⊆ G is open. As G is étale, s(V ) is open.
We are thus nished by proving that (3.13)

Ψ -1 (V ) = [s, X s * ∩ φ -1 (s(σ(s) ∩ V ))] : s ∈ S . If [s, x] ∈ Ψ -1 (V ), then φ(x) = s(Ψ[s, x]) ∈ s(σ(s) ∩ V ) so [s,
x] belongs to the right-hand side of (3.13).

Conversely, if [s, x] belongs to the right-hand side of (3.13), then there is an arrow

γ in σ(s) ∩ V whose source is φ(x). By denition of Ψ, we have Ψ[s, x] = γ ∈ V , so [s, x] ∈ Ψ -1 (V ).
Example 3.14. Following [START_REF] Alan | Groupoids, inverse semigroups, and their operator algebras[END_REF], a localization consists of a global action θ = ({X s } s∈S , {θ s } s∈S ) of an inverse semigroup S on a topological space X such that {X s : s ∈ S} is a basis for the topology of S. The groupoid of germs in the sense of Paterson (see [START_REF] Alan | Groupoids, inverse semigroups, and their operator algebras[END_REF]) coincides with the denition above of groupoids of germs.

Example 3.15. Let X be a topological space. The canonical action of I(X) on X is the action τ given by τ φ = φ for all φ ∈ I(X). A pseudogroup on X is a sub-inverse semigroup of I(X) whose elements are homeomorphisms between open subsets of X.

Let B be a basis for the topology of X, and for each B ∈ B consider its identity function id B : B → B.

Given a pseudogroup G on X, let GB be the sub-inverse semigroup of I(X) generated by G ∪{id B : B ∈ B}, which is again a pseudogroup on X, and in fact the canonical action of GB on X is a localization.

The groupoid of germs in the sense of Renault (see [START_REF]Cartan subalgebras in C * -algebras[END_REF]) of G coincides with the groupoid of germs GB X dened above.

Example 3.16 (Transformation groupoids). In the case that S is a discrete group, the equivalence relation ∼ on S * X is trivial and the topology is the product topology, so S X is (isomorphic to) the usual transformation groupoid.

Example 3.17 (Maximal group image). Suppose that X = {x} is a one-point space on which S acts trivially. Then S X is a groupoid whose unit space is X, a singleton, that is, S X is a group. The universal property of S X implies that S X satises the universal property of the maximal group image G(S) of S (see [START_REF] Alan | Groupoids, inverse semigroups, and their operator algebras[END_REF] or Section 4), so S X is isomorphic to G(S).

Example 3.18 (Restricted product groupoid). Let X = E(S) with the discrete topology, and let θ = ({X s } s∈S , {θ s } s∈S ) be the Munn representation of S (see [START_REF] Walter | Fundamental inverse semigroups[END_REF]): X s = {e ∈ E(S) : e ≤ ss * } and θ s (e) = ses * for all e ∈ X s * .

From S we can construct the restricted product groupoid (S, •), which is the same as S but the product s • t = st is dened only when s * s = tt * . (See [START_REF] Mark | Inverse semigroups[END_REF] for more details.)

Then S E(S) is a discrete groupoid, and the map S E(S)

→ (S, •), [s, e] → se is an isomorphism from S E(S) to (S, •), with inverse s → [s, s * s].
Example 3.19 ([53]). Let S = N ∪ {∞, z}, with product given, for m, n ∈ N,

nm = min(n, m), n∞ = ∞n = nz = zn = n, z∞ = ∞z = z and zz = ∞∞ = ∞.
In other words, S is the inverse semigroup obtained by adjoining the lattice N to the group {∞, z} of order 2 (where ∞ is the unit), in a way that every element of N is smaller than z and ∞. Let X = E(S) = N ∪ {∞}, seen as the one-point compactication of the natural numbers, and θ the Munn representation of S, so that S X = (S, •), however with the topology whose open sets are either conite or contained in N. In particular, S X is not Hausdor.

Example 3.20. Every étale groupoid is isomorphic to a groupoid of germs. Indeed, let G be an étale groupoid, and S any subsemigroup of G op which covers G (i.e., G = A∈S A). We let S act on G (0) by the restriction of the canonical action of G op on G (0) (as Example 2.11). Then the map Φ :

S G (0) → G, [A, x] → s | -1 A (x)
, is a surjective homomorphism of topological groupoids. Moreover, Φ is injective if and only if S forms a basis for some topology on G (see [START_REF]Inverse semigroups and combinatorial C * -algebras[END_REF]Proposition 5.4]).

In particular, if G is an ample groupoid, and γ is the canonical action of G a on G (0) , then the groupoid of germs G a G (0) is (canonically) isomorphic to G.

For the results in Section 8 we will need to consider partial actions with Hausdor groupoids of germs.

Let us mention conditions on inverse semigroups which guarantee that groupoids of germs are Hausdor.

Denition 3.21. A poset (L, ≤) is a (1) (meet-)weak semilattice if for all s, t ∈ L there exists a nite (possibly empty) subset F ⊆ L such that {x ∈ L : x ≤ s and x ≤ t} = f ∈F {x ∈ L : x ≤ f } .
(2) (meet-)semilattice if every pair of elements s, t ∈ L admits a meet, that is, s ∧ t = inf {s, t} exists. Example 3.22. If G is an étale groupoid, then G op is a semilattice. If G is an ample Hausdor groupoid, then G a is a semilattice. In either of these cases, the meets are given by intersection:

U ∧ V = U ∩ V .
Example 3.23. Every E-unitary inverse semigroup S (see Section 4) is a weak semilattice: If s, t ∈ S do not have any common lower bound, F = ∅ in satises Denition 3.211. If s, t have some common lower bound then they are compatible, and s ∧ t = st * t, so instead we take F = {st * t}.

More generally, every E * -unitary inverse semigroup is a semilattice (see [START_REF]Socity and other dynamical aspects of groupoids and inverse semigroups[END_REF]Example 4.5.4]).

The following relation between inverse semigroups which are weak semilattices and the topology of their groupoids of germs can be proven just as in [START_REF]A groupoid approach to discrete inverse semigroup algebras[END_REF]Theorem 5.17]. Proposition 3.24 ([53,Theorem 5.17]). An inverse semigroup S is a weak semilattice if and only if for any partial action θ = ({X s } s∈S , {θ s } s∈S ) of S on a Hausdor space X such that X s is clopen for all s ∈ S, the groupoid of germs S X is Hausdor.

In particular, if S is a weak semilattice and X is zero-dimensional, then the groupoid of germs S X is an ample Hausdor groupoid.

Remark 3.25. The hypothesis that the domains of the partial action are clopen is necessary. For example, if G is a non-Hausdor ample groupoid, then G op is still a semilattice, however, as in Example 3.20, the groupoid of germs G op G (0) ∼ = G is not Hausdor.

Partial actions from associated groups and inverse semigroups

We will now describe how to construct partial actions of groups from actions of inverse semigroups and vice-versa. The class of inverse semigroups which allows us to do this in a more precise manner is that of E-unitary inverse semigroups.

To u ∈ S such that u ≤ s, t.

Alternatively, s ∼ t if and only if there exists e ∈ E(S) such that se = te. From this and the fact that the order of S is preserved under products and inverses, it follows that ∼ is in fact a congruence, so we endow S/∼ with the quotient semigroup structure. Given s ∈ S, we denote by [s] Recall that an inverse semigroup S is E-unitary if whenever e, s ∈ S, e ≤ s and e ∈ E(S), we have s ∈ E(S) as well. We rst reword the E-unitary property in terms of compatibility of elements. Two elements s, t of an inverse semigroup S are compatible if s * t and st * are idempotents. In this case, the set {s, t} has inmum s ∧ t = inf {s, t} = st * t = ts * s. S is E-unitary if and only if s, t, u ∈ S and u ≤ s, t implies that s and t are compatible.

We will now be interested in relating partial actions of inverse semigroups and partial actions of their maximal group images. A version this theorem has been proven in [START_REF] Steinberg | Inverse semigroup homomorphisms via partial group actions[END_REF]Lemma 3.8] when considering global actions of inverse semigroups. The next theorem is a specic instance of [START_REF] Khrypchenko | Partial actions and an embedding theorem for inverse semigroups[END_REF]Lemma 2.2], where the author in fact considers a strictly weaker notion of partial action namely, condition 2.2(iii) is not required. Note that this condition is trivial when considering partial actions of groups, and thus we may apply [33, Lemma of G(S) on X such that for all s ∈ S,

(i) X γ = [s]=γ X s for all γ ∈ G(S); (ii) θ [s] (x) = θ(x) for all (s, x) ∈ S * X;
(in other words, θ γ is the join of {θ s : [s] = γ} in I(X), which is commonly denoted by [s]=γ θ s ). Remark 4.8. If one allows degenerate partial actions, then item (i) implies that θ is non-degenerate if and

only if θ is non-degenerate.
A version of the next theorem has been proven in [START_REF] Milan | On inverse semigroup C * -algebras and crossed products[END_REF], when considering the canonical action of S on the spectrum of its idempotent set E(S). We prove the result for general partial actions of inverse semigroups on arbitrary topological spaces. Proposition 4.9. Let θ be a partial action of an E-unitary group S on a space X and θ be the induced

action on G(S). Then S θ X ∼ = G(S) θ X Proof. Consider the map [s, x] → ([s], x),
s ∧ t). Since x ∈ X s * ∩ X t * ⊆ X s * s ∩ X t * t ⊆ X s * st * t we conclude that [s, x] = [t, y].
The two previous propositions describe a strong relationship between partial actions of an E-unitary inverse semigroup and partial actions of the associated group. The other direction initially reads as follows:

How to associate, to a group G, an inverse semigroup S together with a map G → S such that every partial action of G factors through a partial action of S? The obvious answer would be S = G, so instead we look for global actions of our semigroup S. This is the content of the paper [START_REF]Partial actions of groups and actions of inverse semigroups[END_REF]: Given a group G, let S(G) be the universal semigroup generated by symbols of the form [t], where t ∈ G, modulo the relations

(i) [s -1 ][s][t] = [s -1 ][st]; (ii) [s][t][t -1 ] = [st][t -1 ]; (iii) [s][1] = [s]; (iv) [1][s] = [s];
Exel proved that S(G) is an inverse semigroup with unit [START_REF] Abadie | On partial actions and groupoids[END_REF] (see [START_REF]Partial actions of groups and actions of inverse semigroups[END_REF]Theorem 3.4]). We will describe all the necessary properties of S(G) that we will need. For every g ∈ G, the inverse of

[g] is [g -1 ]. Let us denote g = [g][g -1 ].
By [20, Proposition 2.5 and 3.2], for each γ ∈ S(G), there is a unique n ≥ 0 and distinct elements r 1 , . . . , r n , g ∈ G such that

(1) γ = r1 • • • rn [g], (if n = 0, this is simply [g]), and (2) r i = 1 for all i. We call such a decomposition γ = r1 • • • rn [g] the standard form of γ, which is unique up to the order of r 1 , . . . , r n . Moreover, given g, r ∈ G, we have [g] r = gr [g]. Thus, for γ = r1 • • • rn [g] ∈ S(G), the inverse of γ is written in standard form as γ * = [g -1 ] rn • • • r1 = g -1 rn • • • g -1 r1 [g -1 ],
The idempotents of S(G) are the elements of the form = r1 • • • rn [START_REF] Abadie | On partial actions and groupoids[END_REF].

For any group G the inverse semigroup associated S(G) is E-unitary ([20, Remark 3.5]). Indeed, suppose γ ∈ S(G), ∈ E(S(G)) and ≤ γ. Writing γ and in standard form, we obtain

γ = s1 • • • sn [s] and = e1 • • • em [1]. Since = γ and [1] is a unit of S(G), we obtain e1 • • • em [1] = = γ = e1 • • • em s1 • • • sn [s].
From the uniqueness of the standard form of we conclude that s = 1 and γ is an idempotent. The main result of [START_REF]Partial actions of groups and actions of inverse semigroups[END_REF] is the following property of the semigroup S(G). Although it is proven in principle only for partial on discrete sets, the same proof applies in the topological setting. Proposition 4.10 ([20, Theorem 4.2.]). Let θ = {X s } s∈S , {θ s } s∈S be a topological partial action of a group G on a space X. Then there is a unique topological action θ of S(G) on X such that θ [g] = θ g , for all g ∈ G.

Proposition 4.11. Let G be a group and S(G) the universal semigroup of G. Then the map G → G(S(G)),

g → [[g]], is an isomorphism. Proof. First note that for all s, t ∈ G, [s][t] = [s][t][t -1 [t] = [st] t , Thus the map G → S(G), g → [g], is a partial homomorphism, and the map S(G) → G(S(G)), α → [α], is a homomorphism. So g → [[g]] is a partial homomorphism between groups, hence a homomorphism. Given α ∈ S(G), since α = s1 • • • sn [s] for certain s, s 1 , . . . , s n we get [α] = [[s]], so g → [[g]] is surjective. If [[g]] = 1 = [[1]], then there is an idempotent = e1 • • • en [1] for which e1 • • • en [g] = [g] = [1] = e1 • •
• en and the uniqueness of the standard form implies g = 1.

Corollary 4.12. Let θ be a partial action of an group G on a space X and θ be the induced action of S(G). Note that the construction S → G(S) is functorial, from the category Inv part of inverse semigroups and partial homomorphisms, to the category Grp of groups and their homomorphisms: Given θ : S → T a partial homomorphism, the map s → [θ(s)] is a partial homomorphism from S to the group G(T ), hence a homomorphism, and thus it factors through a homomorphism G(θ) : G(S) → G(T ).

Then G θ X ∼ = S(G) θ X.
Similarly, we have a functor S from Grp to Inv, the subcategory of Inv part consisting of semigroup homomorphisms. It is not hard to see (following the proof of Proposition 4.12) that G • S is naturally isomorphic to the identity of Grp.

Question: Which semigroups are isomorphic to S(G) for some group G? (Note that, up to isomorphism, we need G = G(S).) One condition for such a semigroup is that it satises the ascending chain condition.

The following interesting corollary shows that for such semigroups one can always extend partial actions to actions: Corollary 4.13. Let G be a group, S = S(G) and θ a partial action of S on a set X. Then there exists an action α of S on X such that θ s ≤ α s for all s ∈ S and S θ X = S α X.

Proof. Let

(1) γ be the partial action of G(S) induced by θ;

(2) γ be the composition of γ with the canonical isomorphism G → G(S(G)), g → [[g]];

(3) α be the action of S = S(G) induced by γ ; Then for all s ∈ S, θ

[s] ≤ γ [[s]] = γ s = α [s]
and

S θ X ∼ = G(S) γ X ∼ = G γ X ∼ = S(G) α X = S α X.

Dual partial actions and their crossed products

In [4, Theorem 3.2], Beuter and Gonçalves showed that any Steinberg algebra of a transformation groupoid given by a partial action of a group, A R (G X), is isomorphic to the crossed product A R (X) G. The objective of this section is to present a self-contained proof that generalizes both results above. More precisely, let θ = ({X s } s∈S , {θ s } s∈S ) be a partial action of an inverse semigroup S on a locally compact Hausdor and zero-dimensional space X. Dene, for each s ∈ S,

D s = {f ∈ A R (X) : supp f ⊆ X s } ∼ = A R (X s ),
where the rightmost isomorphism, D s → A R (X s ), is given by restriction: f → f | Xs (the inverse map extends elements of A R (X s ) as zero on X \ X s ). We then dene

α s : D s * → D s f → f • θ s *
(or, more precisely, α s (f ) is the extensions of f • θ s * as zero on X \ X s ).

It is routine to check that α = ({D s } s∈S , {α s } s∈S ) is an algebraic partial action of S on A R (X). In this case, we say that α is the dual partial action of θ.

We will now prove that the Steinberg algebra A R (S θ X) is isomorphic to the crossed product A R (X) α S.

To this, end, we will need a few technical lemmas. Proof. Let A be a compact-open bisection of S X. Since B germ is a basis for S X and A is compact, there exists a nite family {[s i , U i ] :

1 ≤ i ≤ n} in B germ such that each U i is a compact-open subset of X and A = n i=1 [s i , U i ]. Let W 1 = U 1 and for i ≥ 2, let W i = U i \ i-1 j=1 U i . Then the W i are all compact-open subsets of X, and s n i=1 [s i , W i ] = n i=1 W i = n i=1 U i = s n i=1 [s i , U i ] = s(A).
Since the source map is injective on A,

we have A = n i=1 [s i , W i ]. Moreover, if i = j, then s([s i , W i ] ∩ [s j , W j ]) ⊆ s[s i , W i ] ∩ s[s j , W j ] = W i ∩ W j = ∅,
V ⊆ n i=1 X s * i , if n i=1 r i 1 [si,V ] = 0 in A R (S X), then n i=1 r i 1 θs i (V ) δ si = 0 in A R (X) S.
Proof. We proceed by induction on n. The case n = 1 is trivial, for r 1 1 [s1,V ] = 0 implies that either r 1 = 0 or V = ∅, and in either case we have r 1 1 θs 1 (V ) δ s1 = 0.

Assume then that the statement is valid for n, and that we have a sum with n + 1 elements, of the form

(5.3) n i=1 r i 1 [si,V ] + r1 [s,V ] = 0
For every x ∈ V , consider the nite subcolletion F (x) = {i ∈ {1, . . . , n} : [s, x] ∈ [s i , V ]}. Applying both sides of (5.3) on [s, x], we obtain

(5.4) r = - i∈F (x) r i .
Moreover, by denition of F (x), we have [s, x] = [s i , x] for all i ∈ F (x), so there exists t x ∈ S such that (1)

t x ≤ s, s i (i ∈ F (x)); (2) x ∈ X t * x . Fix any compact-open neighbourhood W x ⊆ V ∩ X t * x of x. The collection {W x : x ∈ V } is
an open cover of V , so it admits a nite subcover: There exist x 1 , . . . , x M ∈ V such that {W 1 , . . . , W M } is a cover of V , where W j = W xj . We may, with the same argument as in Lemma 5.1, assume that the W j are pairwise disjoint. Denote also F j = F (x j ) and t j = t xj .

Given any j, we apply Equation (5.4) on x = x j to obtain r = -i∈Fj r i . Thus

r1 [s,V ] = M j=1 r1 [s,Wj ] = - M j=1 i∈Fj r i 1 [s,Wj ] = - M j=1 i∈Fj r i 1 [si,Wj ]
where we used t j ≤ s, s i (i ∈ F j ) and W j ⊆ X t * j in the third equality. Then (5.5)

0 = n i=1 r i 1 [si,V ] + r1 [s,V ] = M j=1   n i=1 r i 1 [si,Wj ] - i∈Fj r i 1 [si,Wj ]   = M j=1   i ∈Fj r i 1 [si,Wj ]   . Now note that supp   i ∈Fj r i 1 [si,Wj ]   ⊆ s -1 (W j ),
and these sets are pairwise disjoint since the W j are pairwise disjoint. Equation (5.5) then implies that for each j,

i ∈Fj

r i 1 [si,Wj ] = 0.
Using the induction hypothesis on Equation (5.6) and summing over j we obtain

0 = M j=1   i ∈Fj r i 1 θs i (Wj ) δ si   = M j=1   n i=1 r i 1 θs i (Wj ) δ si - i∈Fj r i 1 θs i (Wj ) δ si   = n i=1 r i 1 θs i (V ) δ si - M j=1 i∈Fj r i 1 θs i (Wj ) δ si .
(5.7)

Given i ∈ F j , we have t j ≤ s, s i and W j ⊆ X t * j , so we have can again apply Equation (5.4), and the fact that {W j : j = 1, . . . , M } is a partition of V to obtain

M j=1 i∈Fj r i 1 θs i (Wj ) δ si = M j=1 i∈Fj r i 1 θs(Wj ) δ s = M j=1
-r1 θs(Wj ) δ s = -r1 θs(V ) δ s , so the induction step follows from (5.7).

Lemma 5.8.

If n i=1 r i 1 [si,Ui] = 0 in A R (S X), then n i=1 r i 1 θs i (Ui) δ si = 0 in A R (X) S.
Proof. All the subset U 1 , . . . , U n are compact-open, and thus so is U = n i=1 U i . We may nd 2 a partition {V 1 , . . . , V m } (where m ≤ 2 n -1) of U by compact-open subsets of X with the property that each U i is the union of some of the sets V j . In this case, U i ∩ V j = ∅ if and only if V j ⊆ U i . We then have (5.9)

0 = n i=1 r i 1 [si,Ui] = n i=1   j:Vj ⊆Ui r i 1 [si,Vj ]   = m j=1   i:Vj ⊆Ui r i 1 [si,Vj ]   .
For each j, we have supp i:Vj ⊆Ui r i 1 [si,Vj ] ⊆ s -1 (V j ), and these sets are pairwise disjoint. Equation (5.9) implies that for each j, i:Vj ⊆Ui r i 1 [si,Vj ] = 0. By Lemma 5.2, i:Vj ⊆Ui r i 1 θs i (Vj ) δ si = 0 for each j. Summing over j,

0 = m j=1   i:Vj ⊆Ui r i 1 θs i (Vj ) δ si   = n i=1   j:Vj ⊆Ui r i 1 θs i (Vj ) δ si   = n i=1 r i 1 θs i (Ui) δ si .
Theorem 5.10. Let θ = ({X s } s∈S , {θ s } s∈S ) be a partial action of an inverse semigroup S on a locally compact Hausdor and zero-dimensional topological space X. Then the Steinberg algebra of S θ X is isomorphic to the crossed product A R (X) α S, where α = ({D s } s∈S , {α s } s∈S ) is the dual partial action of θ.

Proof. We will use the notation introduced in the denition of crossed product, Denition 2.6.

We will rst show the existence of a homomorphism φ of L to A R (S X) that vanishes on the ideal N , and thus factors through a homomorphism Φ of the quotient L /N = A R (X) α S.

Dene φ :

L → A R (S X) on a generating element f s δ s of L by φ(f s δ s )(a) = f s (r(a)), if a ∈ [s, X s * ] 0, otherwise,
and extend φ linearly to all of L . We rst need check that φ is well-dened, that is, φ(f s δ s ) is a linear combination of characteristic functions of bisections of S X.

We rst write f s = n i=1 r i 1 θs(Ui) for certain r 1 , . . . , r n ∈ R and U i ⊆ X s * compact-open. Then

(5.11)

φ(f s δ s )(a) =      i:a∈[s,X s * ]∩r -1 (θs(Ui)) r i , if a ∈ [s, X s * ] 0, otherwise. As [s, X s * ] ∩ r -1 (θ s (U i )) = [s, U i ], equation (5.11) simply means that (5.12) φ(f s δ s ) = n i=1 r i 1 [s,Ui] whenever f s = n i=1 r i 1 θs(Ui) .
Therefore φ is a well-dened R-module homomorphism from L to A R (S X). Now, we will show that φ is multiplicative. By linearity of φ, it is enough to verify that it is multiplicative on the generators. Notice that supp(φ(f s δ s )) = [s, θ -1 s (supp(f s ))] for every generator f s δ s . Let f s δ s , f t δ t ∈ L and a ∈ S X. There are two possibilities:

Case 1: a ∈ [s, X s * ][t, X t * ] = [st, θ -1 t (X t ∩ X s * )]. Since supp(φ(f s δ s ) * φ(f t δ t )) ⊆ [s, X s * ][t, X t * ], then φ(f s δ s ) * φ(f t δ t )(a) = 0 On the other hand, (f s δ s )(f t δ t ) = α s (α s * (f s )f t )δ st . Since supp(α s * (f s )f t ) = θ -1 s (supp f s ) ∩ supp(f t ) 2
This is a combinatorial fact easily proven by induction, or with the following argument: For any of the 2 n -1 non-zero sequences S ∈ {0, 1} n \ {0}, we set V S = i∈S -1 (1) U i , and disregard any of these sets which are empty.

THE DYNAMICS OF PARTIAL INVERSE SEMIGROUP ACTIONS

then supp(α s (α s * (f s )f t )) = supp(f s ) ∩ θ s (supp(f t ) ∩ X s * )
and this set is contained in X s ∩ θ s (X t ∩ X s * ), which is the domain of the composition θ -1 t • θ -1 s = θ t * • θ s * , and thus in the domain X st of θ (st) * . Hence

θ -1 st (supp(α s (α s * (f s )f t ))) = θ -1 t (θ -1 s (supp f s ) ∩ supp f t ),
and therefore

supp(φ((f s δ s )(f t δ t ))) = [st, θ -1 t (θ -1 s (supp f s ) ∩ supp f t )] which is contained in [st, θ -1 t (X t ∩ X s * )] = [s, X s * ][t, X t * ], so φ((f s δ s )(f t δ t ))(a) = 0 = (φ(f s δ s ) * φ(f t δ t ))(a)
as we expected.

Case 2

: a ∈ [s, X s * ][t, X t * ].
In this case, we can write a = [s, x][t, y] for unique x ∈ X s * and y ∈ X t * with θ t (y

) = x. Since supp(φ(f s δ s )) ⊆ [s, X s * ] then (φ(f s δ s ) * φ(f t δ t ))(a) = b∈r -1 (r(a)) φ(f s δ s )(b)φ(f t δ t )(b -1 a) = φ(f s δ s )[s, x]φ(f t δ t )[t, y] = f s (θ s (x))f t (θ t (y)). On the other hand, a ∈ [s, X s * ][t, X t * ] ⊆ [st, X (st) * ], so φ((f s δ s ) * (f t δ t ))(a) = φ(α s (α s * (f s )f t )δ st )(a) = α s (α s * (f s )f t )(r(a)) = α s (α s * (f s )f t )(θ s (x)) = (α s * (f s )f t )(x) = f s (θ s (x))f t (x) = f s (θ s (x))f t (θ t (y)) = (φ(f s δ s ) * φ(f t δ t ))(a)
as we desired. Now let us prove that φ vanishes on the ideal N . Since φ is a homomorphism, it is enough to show that φ is zero in elements of the form f δ s -f δ t , where s ≤ t and f ∈ D s , because these elements generate N . Let a ∈ S X. Then

• if a ∈ [s, X s * ] then a ∈ [t, X t * ], and φ(f δ s -f δ t )(a) = f (r(a)) -f (r(a)) = 0; • if a ∈ [t, X t * ] \ [s, X s * ] then r(a) / ∈ X s , because r is injective on [t, X t * ], and f (r(a)) = 0 because f ∈ D s . Thus φ(f δ s -f δ t )(a) = 0 -f (r(a)) = 0; • if a / ∈ [t, X t * ] then a / ∈ [s, X s * ] as well, so φ(f δ s -f δ t )(a) = 0 -0 = 0. Therefore, φ factors through the quotient L /N = A R (X) α S to a map Φ : A R (X) α S → A R (S X) satisfying Φ(f δ s ) = φ(f δ s ) whenever f ∈ Ds * .
In order to prove that Φ is bijective, we will show the existence of a map Ψ : A R (S X) → A R (X) α S which is in fact the inverse map of Φ.

By Lemma 5.1, A R (X) is generated, as an R-module, by characteristic functions of compact-open basic bisections (those of the form 1 [s,U ] , where U ⊆ X s is compact-open).

By Lemma 5.1, every element f ∈ A R (S X) may be written as f =

n i=1 r i 1 [si,Ui] ∈ A R (S X), where r 1 , . . . , r n ∈ R and [s 1 , U 1 ], . . . , [s n , U n ] ∈ B germ . Dene Ψ(f ) = Ψ n i=1 r i 1 [si,Ui] := n i=1 r i 1 θs i (Ui) δ si .
By Lemma 5.8, Ψ is well-dened, and clearly additive. To prove that Ψ is a left inverse to Φ, let f s δ s ∈ A R (X) S (where f s ∈ D s ). We already know (Equation (5.12)) that, by writing f s = n i=1 r i 1 θs(Ui) , we have

Ψ(Φ(f s δ s )) = Ψ n i=1 r i 1 [s,Ui] = n i=1 r i 1 θs(Ui) δ s = f s δ s .
Since the elements f s δ s generate A R (X) S as an additive group, we conclude that Ψ • Φ is the identity of A R (X) S. Similarly, the elements of the form r1 [s,U ] (where s ∈ S and U ⊆ X s * is compact-open) generate A R (S X) as an additive group, by Lemma 5.1, and Equation (5.12) again implies

Φ(Ψ(r1 [s,U ] )) = Φ(r1 θs(U ) δ s ) = r1 [s,U ] , therefore Φ • Ψ is the identify of A R (S X).
Remark 5.13. Note that the diagonal subalgebra D R (S X) ∼ = A R ((S X) (0) ) of A R (S X) coincides with span 1 [e,U ] : e ∈ E(S), U ⊆ X e , and so it is mapped, under the isomorphism of the previous theorem, to the diagonal subalgebra span 1 U δ e : e ∈ E(S), U ⊆ X e of the crossed product A R (X) S.

Corollary 5.14. Let G be an ample groupoid. Then the Steinberg algebra A R (G) is isomorphic to the crossed

product A R (G (0) ) µ G op and A R (G (0) ) η G a ,
where µ and η are the dual actions of the canonical actions of G op and G a on G (0) . Proof. By Example 3.20, G is isomorphic to the groupoids of germs G op G (0) and G a G (0) , given by the respective canonical actions of G op and G a on G (0) . The result follows from Theorem 5.10.

It is interesting to note that the crossed products A R (G (0) ) G op and A R (G (0) ) G a arise from global actions, and not simply partial action as in the previous theorem. Further, using Theorem 5.10 and Corollary 5.14 to a groupoid of germs of a partial action, we obtain

A R (X) α S ∼ = A R (S X) ∼ = A R (X) η (S X) a ,
where η is dual to the canonical action of (S X) a on (S X) (0) ∼ = X.

Recovering a topological partial action from a crossed product

In the previous section we realized the Steinberg algebra of an ample groupoid of germs as a crossed product. In this section we will be interested in the opposite direction, that is, to determine which crossed products of the form A R (X) α S can be realized as Steinberg algebras A R (S θ S) in such a way that α is induced by θ. The rst problem we deal with is to nd conditions which allow us to obtain a topological partial action θ of S on X from an algebraic action α.

It is well-know that given a partial action θ = ({X g } g∈G , {θ g } g∈G ) of a group G on a locally compact Hausdor topological space X, there is an associated partial action α = ({D g } g∈G , {α g } g∈G ) of G on the C*-algebra C 0 (X), and conversely, every partial action of a group G on C 0 (X) comes from a partial action of G on X. In [START_REF] Beuter | Partial crossed products as equivalence relation algebras[END_REF], a similar relation is shown at the purely algebraic level. More precisely, let K be a eld and denote by F 0 (X) the algebra of all functions X → K with nite support, endowed with the pointwise operations. Then there is a one-to-one correspondence between the partial actions of a group G on X and the partial actions of G on F 0 (X).

In this section, we will show that the same occurs with partial actions of inverse semigroups. Throughout this section, we will consider that:

• X and Y are locally compact Hausdor and zero-dimensional topological spaces;

• S is an inverse semigroup;

• R is a commutative unital ring; and • A R (X) is the Steinberg algebra of X, i.e., the R-algebra formed by all locally constant, compactly supported, R-valued functions on X, with the pointwise operations.

In order to nd a biunivocal correspondence between partial actions θ = ({X s } s∈S , {θ s } s∈Xs ) of S on X and the dual partial actions α = ({D s } s∈S , {α s } s∈S ) of S in A R (X), we will need a few preliminary results.

Recall that a ring A is said to have local units if, for every nite subset F of A, there exists an idempotent e ∈ A such that r = er = re for each r ∈ F . Such an element e will be referred to as a local unit for the set F . A commutative unital ring R is said to be indecomposable if its only idempotents are 0 and 1 (the trivial ones).

We will prove that, when R is indecomposable, there is a bijection between ideals with local units of A R (X) and open subsets of X. On one hand, if U is an open subset of X, then ( 1)

I(U ) := {f ∈ A R (X) : supp(f ) ⊆ U } ∼ = A R (U )
is an ideal of A R (X) with local units. Indeed, if f 1 , . . . , f n ∈ I(U ) then the characteristic function 1 K , where K = n i=1 supp(f i ), is a local unit for these functions. Moreover, U is compact if and only if I(U ) has an identity, namely, the characteristic function 1 U is its identity. Proposition 6.1. Suppose that R is an indecomposable commutative unital ring. Then the map U → I(U ) is an order isomorphism between the lattices of open subsets of X and of ideals with local units of A R (X). The inverse map is given by I → U(I) := f ∈I supp f . Proof. Let I ⊆ A R (X) be an ideal with local units. Then the inclusion I ⊆ I(U(I)) follows from the denitions of I and U. For the converse, suppose f ∈ A R (X) and supp f ⊆ U(I) = g∈I supp(g). By compactness of supp f , there are f 1 , . . . , f n ∈ I with supp(f ) ⊆ n i=1 supp(f i ). Let e ∈ I be a local unit for f 1 , . . . , f n . Since e is idempotent and R is indecomposable then e = 1 C for some clopen C ⊆ X, and since e is a local unit for f 1 , . . . , f n this means that

n i=1 supp f i ⊆ C. Therefore supp f ⊆ C, and f = f 1 C = f e ∈ I. This proves that I(U(I)) = I.
For the converse, let U ⊆ X be open, so that the inclusion U(I(U )) ⊆ U is also immediate from the denitions of I and U.

If x ∈ U , simply take any compact-open subset V with x ∈ V ⊆ U , so 1 V ∈ I(U ) and x ∈ supp 1 V ⊆ U(I(U )),
which proves that U = U(I(U )).

Corollary 6.2. Suppose that R is an indecomposable commutative unital ring. Then there is an orderisomorphism between unital ideals of A R (X) and compact-open subsets of X.

The following is a particular case of [START_REF] Luiz | Disjoint continuous functions[END_REF]Theorem 3.42]. We sketch its proof for the sake of completeness. Proposition 6.3. Let R be an indecomposable commutative unital ring. Then Γ : A R (Y ) → A R (X) is an R-algebra isomorphism if and only if there exists a (necessarily unique) homeomorphism ϕ :

X → Y such that Γ(f ) = f • ϕ for all f ∈ A R (X).
Proof. Given a commutative ring A, denote by Ω(A) the set of all maximal ideals with local units of A. By Proposition 6.1, the map X x → I(X \ {x}) ∈ Ω(A R (X)) is a bijection, and it is also a homeomorphism when we endow Ω(A R (X)) with the topology generated by all sets of the form

[f ] = {I ∈ Ω(A R (X)) : f ∈ I} (f ∈ A R (X)).
Repeating the same argument with Y in place of X, and using the fact that Γ preserves maximal ideals with local units, we obtain a homeomorphism ϕ :

X ∼ = Ω(A R (X)) → Ω(A R (Y )) ∼ = Y such that supp(f ) = ϕ(supp(Γ(f )) for all f ∈ A R (Y ).
Let x ∈ X be xed, and choose any compact-open neighbourhood U of x and let e = 1 ϕ(U ) ∈ A R (Y ). Then Γ(e) 2 = Γ(e 2 ) = Γ(e), so Γ(e) only takes values 0 and 1 since R is indecomposable. Moreover, ϕ(U ) = supp(e) = ϕ(supp Γ(e)), so supp(Γ(e)) = U , and therefore Γ(e) = 1

U . Now given f ∈ A R (Y ), x r = f (ϕ(x)). We have f (ϕ(x)) = re(ϕ(x)), thus ϕ(x) ∈ supp(f -re) = ϕ(supp(Γ(f ) -rΓ(e))). Therefore x ∈ supp(Γ(f ) -rΓ(e)), so Γ(f )(x) = rΓ(e)(x) = r = f (ϕ(x)).
In particular, from the proposition above, we may conclude that there is bijective anti-homomorphism between the group of all homeomorphism from X to Y , and the group of all R-algebra isomorphisms from A R (Y ) to A R (X), given by

T : Homeo(X, Y ) → Iso(A R (Y ), A R (X)), ϕ → T ϕ ,
where T ϕ (f ) = f • ϕ (compare this with [START_REF] Luiz | Disjoint continuous functions[END_REF]Corollary 3.43]).

Proposition 6.4. Suppose that R is an indecomposable commutative unital ring.

If α = ({D s } s∈G , {α s } s∈S )
is a partial action of S on the algebra A R (X) for which each ideal D s has local units, then there is a partial action θ = ({X s } s∈S , {θ s } s∈S ) of S on X such that α is the dual partial action coming from θ (see Section 5).

Proof. Let α be a partial action of S in A R (X) satisfying the hypotheses above. By Proposition 6.1, for each s ∈ S there is an open subset X s ⊆ X such that

D s = I(X s ) = {f ∈ A R (X) : supp f ⊆ X s } ∼ = A R (X s )
.

By Proposition 6.3, for each isomorphism α s : A R (X s * ) → A R (X s ), there is a unique homeomorphism θ s * : X s → X s * such that α s (f ) = f • θ s * for all f ∈ A R (X s * ) ∼ = D s * .
So we simply let θ = ({X s } s∈S , {θ s } s∈S ), and it is clear that, as long as θ is indeed a partial action, then α is the dual partial action of θ.

To nish the proof we need to we show that θ is indeed a partial action. By its very denition, each X s is open in X and θ s : X s * → X s is a homeomorphism. Non-degeneracy of θ can be proven as follows:

Let x ∈ X and f ∈ A R (X) such that x ∈ supp(f ). Since A R (X) = span s∈S D s , we can write f as f = n i=1 f i for certain elements s i ∈ S and f i ∈ D si ∼ = A R (X si ). In particular, supp f ⊆ n i=1 supp f i ⊆ n i=1
X si and so x ∈ X si for some i. This proves that X = s∈S X s .

So it remains only to prove that s → θ s is a partial homomorphism. Let us verify the conditions of Denition 2.2:

(i) Given s ∈ S, we need to prove that θ s

* = (θ s ) * α s * • α s is the identity on D s * ∼ = A R (X s * ), however for all f ∈ D s * ∼ = A R (X s * ), f • id X s * = α s * • α s (f ) = α s * (f • θ s * ) = f • (θ s * • θ s )
so the uniqueness part of Proposition 6.3 implies that θ s * • θ s = id X s * . Similarly, θ s • θ s * = id Xs , thus θ s * = (θ s ) * . (ii) Let s, t ∈ S. We need to prove that θ s • θ t ≤ θ st . On one hand, note that (under the usual identication A R (U )

∼ = I(U )), f ∈ A R (θ -1 t (X t ∩ X s * )) ⇐⇒ supp f ⊆ θ -1 t (X t ∩ X s * ) ⇐⇒ supp(f • θ t * ) ⊆ X t ∩ X s * ⇐⇒ supp(α t (f )) ⊆ X t ∩ X s * ⇐⇒ α t (f ) ∈ D t ∩ D s * , that is, under the canonical identication, A R (θ -1 t (X t ∩ X s * )) ∼ = α -1 t (D t ∩ D s * ). Since α is a partial action, we obtain A R (θ -1 t (X t ∩ X s * )) ∼ = α -1 t (D t ∩ D s * ) ⊆ D (st) * ∼ = A R (X (st) * ) which implies θ -1 t (X t ∩ X s * ) ⊆ X (st) * . The map α (st) * • α s • α t coincides with the identity on α -1 t (D t ∩ D s * ), however α (st) * (α s (α t (f ))) = α (st) * (α s (f • θ t * )) = α (st) * (f • θ t * • θ s * ) = f • θ t * • θ s * • θ st so again uniqueness in Proposition 6.3 implies that θ t * • θ s * • θ st is the identity on θ -1 t (X t ∩ X * s ). We can conclude that θ s • θ t ≤ θ st . (iii) Suppose s ≤ t in S. Let us prove that θ s ⊆ θ t . We have A R (X s * ) ∼ = D s * ⊆ D t * ∼ = A R (X t * ) so X s * ⊆ X t * . The restriction of α t * to D s coincides with α s * , so for all f ∈ D s ∼ = A R (X s ), f • (θ t | X s * ) = (f • θ t )| X s * = α t * (f )| X s * = α s * (f ) = f • θ s
and again the uniqueness part in Proposition 6.3 implies that θ t | X s * = θ s , so θ s ≤ θ t . Corollary 6.5. Suppose that S is an inverse semigroup, that R is an indecomposable commutative unital ring, and that α = ({D s } s∈S , {α s } s∈S ) is an algebraic partial action of S on A R (X), where each ideal D s has local units. Then A R (X) α S is isomorphic to a Steinberg algebra A R (S θ X) such that α is dual to the topological partial action θ.

Proof. By Proposition 6.4, α is dual to a topological partial action θ of S on X, and Theorem 5.10 implies that A R (S θ X) ∼ = A R (X) α X.

Topologically principal partial actions

In this section our main goal is to introduce topologically principal partial actions of inverse semigroups, which will be used later in our study of continuous orbit equivalence. We then use this notion to describe E-unitary inverse semigroups in terms of the existence of certain topologically principal partial actions.

Let G be a groupoid. The isotropy group at a point

x ∈ G (0) is G x x = {a ∈ G : s(a) = r(a) = x} . Note that G x
x is a group with the operation inherited from G. The isotropy subgroupoid of a groupoid G is the subgroupoid

Iso(G) = x∈G (0) G x x = {a ∈ G : s(a) = r(a)} . Since G (0)
is an open subset of Iso(G), then G (0) ⊆ int(Iso(G)). Following the nomenclature of [START_REF]Cartan subalgebras in C * -algebras[END_REF], a topological groupoid G is eective if the converse inclusion holds, i.e., if G (0) = int(Iso(G)).

A topological groupoid G is topologically principal if the set of points in G (0) with trivial isotropy group is dense in G (0) . By [START_REF]Cartan subalgebras in C * -algebras[END_REF]Proposition 3.6], every Hausdor topologically principal étale groupoid is eective (the Hausdor property is necessary, as the groupoid constructed in Example 3.19 is topologically principal but not eective). Conversely, if G is a second-countable eective (possibly non-Hausdor ) groupoid and G (0) satises the Baire property, then G is topologically principal.

The class of (global) actions of inverse semigroups which correspond to eective groupoids of germs was dened in [START_REF] Exel | The tight groupoid of an inverse semigroup[END_REF]. However, we will be interested in partial actions which correspond to topologically principal groupoids of germs. Since we will not make assumptions of second-countability or the Hausdor property, it is important to distinguish eectiveness and topological principality of groupoids. To avoid confusion on this part, we settle with the nomenclature of [START_REF]Cartan subalgebras in C * -algebras[END_REF].

Moreover, distinct notions of topological freeness for either partial actions of countable groups or global actions of inverse semigroups (see [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF] and [START_REF] Exel | The tight groupoid of an inverse semigroup[END_REF]) have natural generalizations to the context of partial actions of inverse semigroups, however they do not coincide in general.

To avoid any confusion, partial actions which correspond to topologically principal or eective groupoids of germs will be called topologically principal or eective, respectively (so the term topologically free will not be used).

Throughout this section, θ = {X s } s∈S , {θ s } s∈S will always denote a topological partial action of an inverse semigroup S on a topological space X. Denition 7.2 ([24, Denition 4.1]). Let x ∈ X and s ∈ S. We say that [START_REF] Abadie | On partial actions and groupoids[END_REF] x is xed by s if θ s (x) = x;

(2) x is trivially xed by s if there exists e ∈ E(S) such that e ≤ s and x ∈ X e . (In particular, x is xed by s.) The partial action θ is eective if for all s ∈ S, the interior of the set of xed points of s consists of the trivially xed points of s, i.e., int {x ∈ X s * : θ s (x) = x} = {X e : e ∈ E(S) and e ≤ s} .

A proof analogous to that of [START_REF] Exel | The tight groupoid of an inverse semigroup[END_REF]Theorem 4.7] proves that eective partial actions correspond to eective groupoids of germs.

Proposition 7.3. The groupoid of germs S θ X is eective if and only if θ is eective.

If θ is a partial action of an inverse semigroup S on a set X, the subset {s ∈ S : x ∈ X s * } of S will be denoted by S x . Denition 7.4. We denote by Λ(θ) the set of points of X which are trivially xed whenever they are xed, i.e., Λ(θ) = {x ∈ X : for all s ∈ S x , if θ s (x) = x then there exists e ∈ E(S) ∩ S x with e ≤ s} .

We say that θ is topologically principal if Λ(θ) is dense in X.

Similarly to the two descriptions of the germ relation as in Equations (3.1) and (3.2), we can alternatively describe Λ(θ) as (7.5) Λ(θ) = {x ∈ X : for all s, t ∈ S x , if θ s (x) = θ t (x) then there exists u ∈ S x with u ≤ s, t} .

Suppose now that θ is a partial action of S on a discrete space X -that is, a set. As closures and interiors of discrete spaces are trivial, we may rewrite both topological principality and eectiveness of θ as follows: for all (s, x) ∈ S * X, if θ s (x) = x then there exists e ∈ S x ∩ E(S) with e ≤ s. In particular, θ is eective if and only if it is topologically principal, thus we can unambiguously call it free.

More generally, by a free partial action θ of S on a topological space X, we mean a partial action which is free when X is regarded simply as a set. Equivalently, this is to say that Λ(θ) = X.

In the case that G is a group, a partial action θ of G is free if for all x ∈ X (and for all g ∈ G x ), one has that θ g (x) = x implies g = 1, where 1 is the identity of G, which is the usual notion of freeness for partial group actions.

It is interesting to note that freeness of a topological partial action implies that the associated groupoid of germs is Hausdor. However, this is not true for topologically principal partial actions. Proposition 7.6. If the action θ is free, then the groupoid of germs S X is Hausdor.

Proof. Suppose [s, x] = [t, y]. First assume that s[s, x] = s[t, y], that is, x = y. As X is Hausdor, choose disjoint neighbourhoods U and V of x and y in X, respectively. Then s -1 (U ) and s -1 (V ) are disjoint neighbourhoods of [s, x] and [t, y], respectively. Similarly, if r[s, x] = r[t, y], we may nd disjoint neighbourhoods of [s, x] and [t, y], respectively.

We are done if we prove that the two cases above are the only possibilities. Suppose then s[s, x] = s[t, y] and r[s, x] = r[t, y], that is, x = y and θ s (x) = θ t (y) = θ t (x). By freeness of θ, there is u ∈ S, such that u ≤ s, t and x ∈ X u * , which is equivalent to stating [s, x] = [t, x] = [t, y], a contradiction. Example 7.7. As in Example 3.19, let S = N ∪ {∞, z} and θ be the Munn representation of S on X = E(S) = N ∪ {∞}, endowed with the same topology as the one-point compactication of N. This is a topologically principal partial action, since Λ(θ) = N is dense in X, however the associated groupoid of germs S X is not Hausdor.

In the specic setting of topological partial actions of countable groups on locally compact Hausdor and second-countable spaces, [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF] adopts a notion of topological freeness which happens to coincide (in this specic setting) with both eectiveness and topologically principality partial actions (of groups). The following proposition can be proven as in [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF]Lemma 2.4], as an application of Baire's Category Theorem. Proposition 7.8. Suppose that S is countable and that X is locally compact Hausdor. Then the partial action θ of S on X is topologically principal if and only if for all s ∈ S, the set {x ∈ X s * : if θ s (x) = x then there exists e ∈ E(S) ∩ S x with e ≤ s} is dense in X s * .

We will now reword topological principality of a partial action in terms of the groupoid of germs S X.

Proposition 7.9. The groupoid of germs S X is topologically principal if and only if θ is topologically principal.

Proof. As usual, we may assume the action θ is non-degenerate and identify X with (S X) (0) . Then it is enough to prove that, under this identication, Λ(θ) is the set of points of X with trivial isotropy, i.e., Λ(θ) = {x ∈ X : (S X) x x = {x}} .

Let x ∈ X be given. First suppose x ∈ Λ(θ) and [s, x] ∈ (S X) x x . This means that x = r[s, x] = θ s (x), so there is e ∈ E(S) ∩ S x , e ≤ s, which implies [s, x] = [e, x] x.

Conversely suppose (S X) x x = {x} and let s ∈ S x with θ s (x) = x. This means that [s, x] ∈ (S X) x

x , and so [s, x] x [e, x] for some idempotent e ∈ S x . By denition of the groupoid of germs, we can nd another idempotent f ∈ S x with se = ef , so in particular ef is an idempotent, ef ≤ s, and x ∈ X ef . This proves x ∈ Λ(θ).

We nish this section by describing how E-unitary inverse semigroups can be characterized in terms of their partial actions.

Proposition 7.10. Suppose that S is E-unitary and that θ = {X γ } γ∈G(S) , θ γ γ∈G(S)

is the unique partial action of G(S) on X given by Theorem 4.7. Then θ is topologically principal if and only if θ is topologically principal.

Proof. We will prove that Λ(θ) = Λ( θ). Suppose that x ∈ Λ(θ), and that s ∈ S is such that x = θ [s] (x) = θ s (x). As x ∈ Λ(θ), there exists e ∈ E(S) ∩ S x with e ≤ s. In particular, se = e, so [s] = [e] = 1, the unit of G(S). This proves that Λ(θ) ⊆ Λ( θ).

Conversely, assume x ∈ Λ( θ), and that s ∈ S x is such that x = θ s (x) = θ [s] (x). This implies that [s] = 1 = [s * s], so there is an idempotent e ∈ E(S) with se = s * se. In particular, s ≥ s * se, which is idempotent, so s is itself an idempotent because S is E-unitary. It follows that s * s ∈ E(S) ∩ S x , and s * s = s. This proves that Λ( θ) ⊆ Λ(θ). Lemma 7.11. Suppose that θ is topologically principal, and that X s = ∅ for all s ∈ S. Then E(S) = {s ∈ S : θ s is idempotent}.

Proof. Suppose θ s is an idempotent. Since X s * = ∅, choose any x ∈ X s * ∩ Λ(θ). Then θ s (x) = x, which implies that there is some e ∈ E(S) with e ≤ s, so s is idempotent because S is E-unitary. Lemma 7.12. Let S be an inverse semigroup and θ = {X s } s∈S , {θ s } s∈S be a partial action of S a space X such that (i) θ factors through G(S) there is a partial action θ = X

[s] [s]∈G(S) , θ [s] [s]∈G(S)
such that

θ [s] (x) = θ s (x) for all x ∈ X s * ; (ii) E(S) = {s ∈ S : θ s is idempotent}. Then S is E-unitary.
Proof. Suppose e ∈ E(S), e ≤ s. We have 1 = [e] = [s], thus for all x ∈ X s * , θ s (x) = θ [s] (x) = θ 1 (x) = x, so θ s is an idempotent and s is idempotent by (ii).

Given an inverse semigroup S, we will consider the canonical action of S on itself as the action α = {D s } s∈S , {α s } s∈S , where D s = {t ∈ S : t * t ≤ ss * }, and α s (t) = st for t ∈ D s * . (This action is usually considered when one proves the Vagner-Preston theorem.) Theorem 7.13. S is E-unitary if and only if it admits a topologically principal partial action satisfying (i) and (ii) of Lemma 7.12.

Proof. One implication is proven in Lemma 7.12. Assume then that S is E-unitary, and let us prove that the canonical action α of S is free: Suppose st = t, where tt * ≤ s * s. Then s ≥ stt * = tt * , which is idempotent, so s is itself an idempotent. This clearly implies that the action α is free. Condition (i) follows from Theorem 4.7, and condition (ii) from Lemma 7.11.

In fact, condition (ii) of Lemma 7.12 is always satised by the canonical action α of an inverse semigroup S on itself: if α s is idempotent, then s = ss * s = α s (s * s) = s * s is idempotent. We thus obtain: Corollary 7.14. S is E-unitary if and only if the canonical action of S factors through G(S).

Continuous Orbit Equivalence

In [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF], Li characterized continuous orbit equivalence of topologically free partial actions of countable groups on second-countable, locally compact Hausdor spaces in terms of diagonal-preserving isomorphisms of the associated C*-crossed products. In this section, we will extend the notion of continuous orbit equivalence to partial actions of inverse semigroups and characterize orbit equivalence of topologically principal systems in terms of diagonal-preserving isomorphisms of the associated crossed products.

Throughout this section, θ = {X s } s∈S , {θ s } s∈S and γ = {Y t } t∈T , {γ t } t∈T will always denote topo- logical partial actions of inverse semigroup S and T on topological spaces X and Y , respectively. Recall that S * X = {(s, x) ∈ S × X : s ∈ S and x ∈ X s * } (and similarly for T * Y ). We regard S and T as discrete topological spaces. such that for all x ∈ X, s ∈ S x , y ∈ Y and t ∈ T y , (i) ϕ(θ s (x)) = γ a(s,x) (ϕ(x));

(ii) ϕ -1 (γ t (y)) = θ b(t,y) (ϕ -1 (y)). Implicitly, we require that a(g, x) ∈ T ϕ(x) and b(t, y) ∈ S ϕ -1 (y) . We call the triple (ϕ, a, b) a continuous orbit equivalence from θ to γ.

Our next goal is to prove that continuous orbit equivalence of topologically principal partial actions is equivalent to the isomorphism of the respective groupoids of germs. For this, we need to prove some identities related to how the functions a and b above preserve the structure of S and T . Lemma 8.2. Let (ϕ, a, b) be a continuous orbit equivalence from θ to γ. Assume that X and Y are Hausdor.

Then the following implications hold:

(

a) [s 1 , x] = [s 2 , x] ⇒ [a(s 1 , x), ϕ(x)] = [a(s 2 , x), ϕ(x)], for all x ∈ X and s 1 , s 2 ∈ S x . (b) [a(s 1 s 2 , x), ϕ(x)] = [a(s 1 , θ s2 (x))a(s 2 , x), ϕ(x)] for all x ∈ X, s 2 ∈ S x and s 1 ∈ S θs 2 (x) . (c) [b(a(s, x), ϕ(x)), x] = [s, x],
for all x ∈ X and s ∈ S x . Analogous statements hold with (ϕ -1 , b, a) in place of (ϕ, a, b).

Proof.

(a) Let x ∈ X and s 1 , s

2 ∈ S x . Suppose that [s 1 , x] = [s 2 , x]. First, choose s ≤ s 1 , s 2 such that x ∈ X s * . Then choose an open neighbourhood U ⊆ X s * of x ∈ X such that a(s 1 , x) = a(s 1 , x) and a(s 2 , x) = a(s 2 , x) whenever x ∈ U.
Then for all x ∈ U ∩ ϕ -1 (Λ(γ)) and for i = 1, 2, we have [s

i , x] = [s, x], so γ a(si,x) (ϕ(x) = γ a(si, x) (ϕ(x)) = ϕ(θ si (x)) = ϕ(r[s i , x]) = ϕ(r[s, x]).
It follows that γ a(s1,x) (ϕ(x)) = γ a(s2,x) (ϕ(x)). As ϕ(x) ∈ Λ(γ), the description of Λ(γ) as in Equation (7.5) implies that

(8.3) [a(s 1 , x), ϕ(x)] = [a(s 2 , x), ϕ(x)] for all x ∈ U ∩ ϕ -1 (Λ(γ)).
In particular, [a(s i , x), ϕ(x)] and [a(s i , x), ϕ(x)] belong to the bisection [a(s 1 , x), ϕ(U )], which is

Hausdor.

Since γ is topologically principal, Λ(γ) is dense in Y , so U ∩ ϕ -1 (Λ(γ)) is dense in U and therefore we may take the limit x → x in Equation (8.3) and conclude that [a(s 1 , x), ϕ(x)] = [a(s 2 , x), ϕ(x)], limits are unique in Hausdor spaces.

(b) Choose an open neighbourhood U of x ∈ X such that a(s 1 s 2 , x) = a(s 1 s 2 , x), a(s 1 , θ s2 (x)) = a(s 1 , θ s2 (x)) and a(s 2 , x) = a(s 2 , x) for all x ∈ U . Then for all x ∈ U ∩ ϕ -1 (Λ(γ)) γ a(s1s2,x) (ϕ(x)) = ϕ(θ s1s2 (x)) = ϕ(θ s1 (θ s2 (x)) = γ a(s1,θs 2 (x)) (ϕ(θ s2 (x))) = γ a(s1,θs 2 (x)) (γ a(s2,x) (ϕ(x))) = γ a(s1,θs 2 (x))a(s2,x) (ϕ(x))
so, the same way as in item (a), the given property of U and the denition of Λ(γ

) imply that [a(s 1 s 2 , x), ϕ(x)] = [a(s 1 , θ s2 (x))a(s 2 , x), ϕ(x)]. Since ϕ -1 (Λ(γ)) ∩ U is dense in the Hausdor space U , we conclude that [a(s 1 s 2 , x), ϕ(x)] = [a(s 1 , θ s2 (x))a(s 2 , x), ϕ (x
)] by taking the limit x → x.

(c) Similarly to the previous items, take neighbourhoods U of x and V of ϕ(x) such that a(s, x) = a(s, x) and b(a(s, x), ỹ) = b(a(s, x), ϕ(x))

whenever x ∈ U and ỹ ∈ V . Then for all x ∈ U ∩ ϕ -1 (V ) ∩ Λ(θ), θ b(a(s,x),ϕ(x)) (x) = ϕ -1 (γ a(s,x) (ϕ(x))) = ϕ -1 (ϕ(θ s (x))) = θ s (x)
so the properties of U , V and Λ(θ) yield [b(a(s, x), ϕ(x)), x] = [s, x] and again taking x → x gives us the desired result.

Theorem 8.4. Suppose that θ and γ are topologically principal, continuously orbit equivalent partial actions, and that X and Y are Hausdor. Then S X and T Y are isomorphic as topological groupoids.

Proof. Let (ϕ, a, b) be a continuous orbit equivalence from θ to γ (as in Denition 8.1). Then the map

Φ : S X → T Y, Φ[s, x] = [a(s, x), ϕ(x)]
is a continuous groupoid homomorphism. Indeed, by Lemma 8.2(a), Φ is well-dened, and item (b) of that lemma implies that Φ is a homomorphism. As a and ϕ are continuous, it follows that Φ is continuous.

Similarly, the map

Ψ : T Y → S X, Ψ[t, y] = [b(t, y), ϕ -1 (y)]
is a continuous groupoid homomorphism, and Ψ is a left inverse to Φ by Lemma 8.2(c). The same arguments with (ϕ -1 , b, a) in place of (ϕ, a, b) prove that it is also a right inverse.

We will now be interested in constructing an orbit equivalence for two actions from an isomorphism of the corresponding groupoids of germs. Note that in general the continuous maps a and b in the denition of continuous orbit equivalence take values in discrete spaces (namely, the corresponding semigroups), and so X and Y are required to have suciently many clopen sets in order for a continuous orbit equivalence between the corresponding partial actions to exist. Thus we concentrate on spaces which have suciently many clopen sets and partial actions which respect this structure.

The required property for the topological spaces that we will need to consider is ultraparacompactness, which is a stronger version of zero-dimensionality and covers most cases of interest (namely locally compact Hausdor and zero-dimensional spaces which are also second countable or compact; see Example 8.6).

We extend the source map to E ∪ E ∞ , and the range map to E as follows:

If v ∈ E 0 is a vertex, then s(v) = r(v) = v; If µ = µ 1 • • • is a path (nite or innite) of length |µ| ≥ 1, then s(µ) = s(µ 1 ); If µ = µ 1 • • • µ |µ| is nite, we set r(µ) = r(µ |µ| ).
Paths can be concatenated if their range and source agree, as long as we take the proper care with vertices: if v is a vertex and µ ∈ E is such that r(µ) = v, then we specify that µv = µ, and similarly

if ν ∈ E ∪ E ∞ is such that s(ρ) = v, we set vρ = ρ. If µ = µ 1 • • • µ |µ| ∈ E and ν = ν 1 • • • ∈ E ∪ E ∞ are paths of length ≥ 1 with r(µ) = s(ν), we set µν = µ 1 • • • µ |µ| ν 1 • • • .
Note that we always have |µν| = |µ| + |ν| whenever the concatenation µν is dened.

A vertex v is called a sink if s -1 (v) = ∅ and it is called an innite emitter if |s -1 (v)| = ∞. If v ∈ E 0
is either a sink or an innite emitter then it is called singular.

The boundary path space of E is dened as

∂E := E ∞ ∪ {µ ∈ E : r(µ) is singular} .
For a nite path µ ∈ E , we dene the cylinder set

Z(µ) = {µx | x ∈ ∂E and r(µ) = s(x)} ⊆ ∂E,
and for a nite set F ⊆ s -1 (r(µ)) (possibly empty), we dene the generalised cylinder set Graph semigroup. We will associate an inverse semigroup S E to the graph E = (E 0 , E 1 , s, r).

Z(µ, F ) = Z(µ) \ e∈F Z(µe) = {µx | x ∈ ∂E, x 1 / ∈ F and r(µ) = s(x)}.
Let

S E = {(µ, ν) | µ, ν ∈ E and r(µ) = r(ν)} ∪ {0}.
The product is determined by setting 0 as a zero (absorbing) element, and

(9.2) (µ, ν)(ζ, η) =      (µ, ηγ), if ν = ζγ for some γ ∈ E (µγ, η), if ζ = νγ for some γ ∈ E 0, otherwise.
This operation makes S E into an inverse semigroup, with the inverse given by (µ, ν) * = (ν, µ) and 0 * = 0 (see [START_REF]Graph inverse semigroups, groupoids and their C * -algebras[END_REF]Proposition 3.1] for a proof ). The set E(S E ) of idempotents of S E coincides with the set of all pairs (µ, µ), where µ ∈ E , and the zero element 0.

Notice that the product of two pairs (µ, ν), (ζ, η) is non-zero if, and only if, ν is an initial segment of ζ or vice-versa. In this case, we say that ν and ζ are comparable. It is easy to see that if (µ, ν) ≤ (ζ, η) if and only if there is γ ∈ E such that (µ, ν) = (ζγ, ηγ). From this, it follows that S E is a semilattice (actually, an E * -unitary inverse semigroup).

We will now describe the canonical action of S E on the boundary path space ∂E. Given (µ, ν) ∈ S E \ {0} Since S E is a semilattice and the subsets Z(µ) are all compact and in particular clopen in ∂E, then the groupoid of germs S E ∂E is Hausdor by Proposition 3.24, and ample since ∂E is locally compact Hausdor and zero-dimensional. and it is not hard to see that, conversely, items (i)'' and (ii)'' imply (i)-(iii).

Finally, items (i)'' and (ii)'' clearly imply that ((µ 1 , ν 1 ), x 1 ) ∼ ((µ 2 , ν 2 ), x 2 ), and the converse is also true because S E is E * -unitary. Therefore ψ factors through a groupoid isomorphism Ψ : S E ∂E → G E . To verify that Ψ is a homeomorphism, note that a basic (nonempty) open subset of S E ∂E has the form [(µ, ν), Z(νη, F )], where (µ, ν) ∈ S E , η ∈ s -1 (r(ν)) and F is a nite subset of r -1 (s(η)). A loop or cycle in a graph E is a nite path y ∈ E such that |y| ≥ 1 and s(y) = r(y). An exit of a loop y = y 1 • • • y |y| (where y i ∈ E 1 ) is an edge e for which there is i such that s(e) = s(y i ) and e = y i . The graph E is said to satisfy Condition (L) if every loop has an exit. Denition 9.7 ([7,Denition 3.1]). Two directed graphs E = (E 0 , E 1 , r, s) and F = (F 0 , F 1 , r, s) are continuously orbit equivalent if there exists a homeomorphism ϕ : ∂E → ∂F together with continuous maps k, l : ∂E ≥1 → N and k , l : ∂F ≥1 → N such that Here, σ E and σ F denote the shift maps on E and F , respectively.

The following is an analogue of [START_REF] Brownlowe | Graph algebras and orbit equivalence[END_REF]Proposition 2.3]. We provide a simple proof for completeness. Proposition 9.10. Let E = (E 0 , E 1 , r, s) be a directed graph. Then E satises Condition (L) if and only if the canonical action θ of S E on ∂E is topologically principal (or equivalently, G E is topologically principal).

Proof. Let us say that an element x ∈ ∂E is cyclic if there exists x ∈ E with |x | ≥ 1 such that x = x x, or equivalently x = x x x • • • , and that x is periodic if x = νy for some ν ∈ E and some cyclic y.

First suppose that E satises Condition (L). Consider the set X = (E ∩∂E)∪{x ∈ E ∞ : x is not periodic}. Condition (L) implies that X is dense in ∂E. We are done by proving that X ⊆ Λ(θ). Suppose (µ, ν) ∈ S E and x = νy ∈ Z(ν) is such that θ (µ,ν) (x) = x. Let us prove that µ = ν. We have (9.11) µy = θ (µ,ν) (x) = x = νy.

It follows that µ and ν are comparable, so to prove that µ = ν it suces to prove that |µ| = |ν|. Without loss of generality, let us assume that µ = νµ for some µ . From (9.11) we obtain y = µ y. However, y is not cyclic, since x is not periodic, so |µ | = 0, and |µ| = |νµ | = |ν|. We conclude that θ is topologically principal.

Conversely, suppose E does not satisfy Condition (L), and let y be any loop in E without exit. The element x = yyy • • • is isolated in ∂E, because Z(y) = {x}, and θ (y,yy) (x) = x. However, the only idempotent in S E which is smaller than (y, yy) is the zero, and θ 0 is the empty function, thus Z(y) ∩ Λ(θ) = ∅. This proves that Λ(θ) is not dense in ∂E, therefore θ is not topologically principal.

We will now compare continuous orbit equivalence of graphs and continuous orbit equivalence of the canonical action of the associated semigroups. The following is analogue to [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF]Lemma 3.8], but we do not require that the graphs satisfy Condition (L). Proposition 9.12. Let E = (E 0 , E 1 , s, r) and F = (F 0 , F 1 , s, r) be directed graphs. Then E and F are continuously orbit equivalent if and only if the canonical actions θ E and θ F associated to E and F are continuously orbit equivalent.

Proof. Assume that (ϕ, a, b) is a continuous orbit equivalence between θ E and θ F . Given x ∈ ∂E ≥1 , let us denote by x 1 ∈ E 1 the rst edge of x (i.e., x = x 1 y for some y ∈ ∂E). The map x → x 1 is locally constant on ∂E ≥1 namely, it is the constant map x → e on Z(e) for each e ∈ E 1 , and Z(e) : e ∈ E 1 is a partition of ∂E ≥1 .

Let α, β : ∂E ≥1 → S E be functions such that a((r(x 1 ), x 1 ), x) = (α(x), β(x)) for all x ∈ ∂E ≥1 and so (9.8) holds. To prove (9.9), k and l are dened in a similar way, using b.

Conversely, suppose ϕ : ∂E → ∂F is a homeomorphism and that there are maps k, l : ∂E ≥1 → N satisfying, for all x ∈ ∂E ≥1 , (9.13) σ k(x) F (ϕ(σ E (x))) = σ l(x) F (ϕ(x)). We must show that there is a continuous function a : S E * ∂E → S F such that (9.14) ϕ(θ E (µ,ν) (x)) = θ F a(µ,ν,x) (ϕ(x)), for all (µ, ν) ∈ S E and x ∈ Z E (ν). By Lemma 8.8, it is sucient to prove that for all (µ, ν) ∈ S E and for all x ∈ Z(ν), there exists an open set U containing x and (α, β) ∈ S F such that for all x ∈ U , ϕ(θ E (µ,ν) ( x)) = θ F (α,β) (ϕ( x)).

Let us separate the proof in cases:

(1) Assume that |µ| = |ν| = 0 (which implies that µ = ν).

In this case, we simply take U = Z E (ν) ∩ ϕ -1 (Z F (s(ϕ(x)))). Then for all x ∈ U , ϕ(θ E (µ,ν) ( x)) = ϕ( x) = θ F (s(ϕ(x)),s(ϕ(x))) (ϕ( x)), so we are done.

(2) Assume that |µ| = 0 and |ν| = 1.

Let K = k(x) and L = l(x). For all x ∈ Z E (ν) ⊆ ∂E ≥1 , we have θ E (µ,ν) ( x) = σ E ( x)

Let U 1 = Z E (ν) ∩ k -1 (K) ∩ l -1 (L). Then for all x ∈ U 1 , Equation (9.13) implies that

  the equivalence class of s with respect to the relation (4.1). The following proposition is a particular case of Proposition 3.4 and Theorem 3.11 (see Example 3.17).

  Proposition 4.2 ([45, Proposition 2.1.2]). Let S an inverse semigroup. The quotient G(S) := S/∼is a group. Furthermore, G(S) is the maximal group homomorphic image of S in the sense that if ψ : S → G is a homomorphism and G is a group, then ψ factors through G(S).

Example 4 . 3 .

 43 If G is a group then G(G) is isomorphic to G. Example 4.4. If L is a (meet-)semilattice then G(L) = {1} is the trivial group.Example 4.5. If S is an inverse semigroup with a zero, then G(S) = {1} is the trivial group.

Lemma 4 . 6 (

 46 [START_REF] Mark | Inverse semigroups[END_REF] Theorem 2.4.6]).

2. 2 ]

 2 without problems.Theorem 4.7 ([33, Remark 2.3]). Let θ = {X s } s∈S , {θ} s∈S be a partial action of an E-unitary inverse semigroup S on a topological space X. Then there is a unique partial action θ = {X γ } γ∈G(S) , θ γ γ∈G(S)

  Proof. Let γ = θ, the partial action of G(S(G)) induced by θ as in Theorem 4.7. Let us prove that for all g ∈ G, θ g = γ [[g]] . From this fact and Proposition 4.11, it follows easily thatG θ X → G(S(G)) γ X, (g, x) → ([[g]], x)is a topological groupoid isomorphism. Proposition 4.9 provides the isomorphism G(S(G)) γ X ∼ = S(G) θ X, so we are done.Let g ∈ G be xed. By denition, γ [[g]] is the supremum of θ s : s ∼ [g] . From the uniqueness of the standard form of each s ∈ S(G), it follows that s ∼ [g] if and only if s ≤ [g], and thus we conclude that γ [[g]] = θ [g] .

Lemma 5 . 1 .

 51 Every compact-open bisection of S X is a disjoint union of compact-open elements of B germ (see Proposition 3.7).

  and thus we have a partition of A by compact-open elements of B germ . Lemma 5.2. For every pair of nite families s 1 , . . . , s n ∈ S and r 1 , . . . , r n ∈ R, and for every compact-open subset

Remark 7 . 1 .

 71 The nomenclature essentially principal has been used to mean either eective or topologically principal (or even slight variations) in dierent works. See [47, Denition II.4.3], [51, Section 2.2] and [24, Denition 4.6(4)].

Denition 8 . 1 .

 81 We say that θ and γ are continuously orbit equivalent if there exist a homeomorphism ϕ : X -→ Y and continuous maps a : S * X -→ T and b : T * Y -→ S

θ

  (µ,ν) : Z(ν) → Z(µ), νx → µx, and θ 0 : ∅ → ∅ the empty map. The verication that the collectionθ = {Z(µ)} (µ,ν)∈S E , θ (µ,ν) (µ,ν)∈S Eis a topological (global) action of S E on ∂E is straightforward, by considering the dierent cases as in Equation (9.2).

  Then

Ψ

  ([(µ, ν), Z(νη, F )]) = (µηx, |µ| -|ν|, νηx) : x ∈ s -1 (r(η)) \ e∈F Z(e) = Z(µη, νη, F ) and these are precisely the basic open subsets of G E . Therefore, Ψ is a homeomorphism.

  σ E (x))) = σ l(x) F (ϕ(x)), for all x ∈ ∂E ≥1 , 1 (σ F (x))) = σ l (y) E (ϕ -1 (y)), for all y ∈ ∂F ≥1 .

  , and dene k(x) = |α(x)| and l(x) = |β(x)|. As a is continuous, then k and l are continuous. Moreover, we haveϕ(σ E (x)) = ϕ(θ E (r(x1),x1) (x)) = θ F (α(x),β(x)) (ϕ(x)),which means that ϕ(σ E (x)) = α(x)y and ϕ(x) = β(x)y, for some y ∈ ∂F . Thus σ

1 )

 1 ν) ( x)) = σ L F (ϕ( x))Equation (9.15) with x = x implies that there exist (α,β) ∈ S F , with |α| = K and |β| = L, such that ϕ(θ E (µ,ν) (x)) = θ F (α,β) (ϕ(x)). Thus setting U = U 1 ∩ ϕ -1 (Z F (ν)) ∩ (ϕ • θ E (µ,ν) ) -1 (Z F (µ)), we obtain Equation (9.14) on U . (3) Assume that |µ| = 0 and |ν| ≥ 1. Write ν = ν 1 • • • ν |ν| , where ν i ∈ E 1 . Notice that (µ, ν) = (µ, ν |ν| )(s(ν |ν| ), ν |ν|-1 ) • • • (s(ν 3 ), ν 2 )(s(ν 2 ), νIn other words, there are elements e 1 , . . . , e |ν| of the form considered in the previous case, such that (µ, ν) = e |ν| • • • e 1 . Applying the previous case, for each k ≥ 1 we may nd a neighbourhood U k of θ e k-1 •••e1 (x) (or simply x in the case k = 1) and an element f k ∈ S F such thatϕ • θ E e k = θ F f k • ϕ

  Indeed, if (s, x) ∼ (t, y) and u ∈ S satises(3.1), then e = u * u satises (3.2). Conversely, if e ∈ E(S) satises (3.2), then u = se satises (3.1).We call the ∼-equivalence class of (s, x) is the germ of s at x, and we denote it by [s, x]. Remark 3.3. If u ≤ s in S and x ∈ X u * , then x ∈ X s * as well and [s, x] = [u, x].

	Proposition 3.4. The relation ∼ is a congruence, and the quotient semigroupoid S X := (S * X)/∼ is a
	groupoid. The inverse of [s, x] ∈ S X is [s

* , θ s (x)].

  z ≤ s, t} . Proposition 3.7. The family B germ of sets [s, U ], where s ∈ S and U ⊆ X s * is open, forms a basis for a topology on S X, which makes it a topological groupoid.

	Proof. By (3.6), B germ is a basis for a topology on S X.

  Proposition 3.9. S X is an étale groupoid, and each basic open set [s, U ] ∈ B germ is an open bisection. Proof. Given [s, U ] ∈ B germ , we have s[s, U ] = [s * s, U ] ∈ B germ , so the source map is open. Moreover, it is injective on [s, U ]. Therefore the source map is locally injective, continuous and open, hence a local homeomorphism, so S X is étale. Similarly, the range map is also injective on [s, U ], which is therefore a bisection.

  In the same paper ([4, Theorem 5.2]), they proved that every Steinberg algebra associated with an ample Hausdor groupoid G, is isomorphic to the crossed product A R (G (0) ) G a . Similarly, in [16, Theorem 2.3.6], Demeneghi proved that any Steinberg algebra of a groupoid of germs associated to an ample global action of an inverse semigroup is isomorphic to a crossed product A R (X) S, and as a consequence obtained the latter result presented by the previous authors (see[START_REF] Demeneghi | The ideal structure of Steinberg algebras[END_REF] Proposition 2.4.3]). However, [4, Theorem 3.2] considers partial (non-global) actions of groups, and thus does not follow from[START_REF] Demeneghi | The ideal structure of Steinberg algebras[END_REF] Theorem 2.3.6].

Acknowledgements. We thank Thierry Giordano for the several useful comments and suggestions, and who worked as both the second named author's PhD supervisor (co-supervised by Vladimir Pestov), and as the rst named author's supervisor during her stay at the University of Ottawa as a Visiting Student Researcher during the Winter Session of 2018.

Preliminaries

Inverse semigroups. A semigroup is a set endowed with an associative binary operation (s, t) → st, called

† Supported by CAPES/Ciência Sem Fronteiras PhD scholarship 012035/2013-00, and by ANR project GAMME (ANR-14-CE25-0004). 1 product.

We refer to [START_REF] Ellis | Extending continuous functions on zero-dimensional spaces[END_REF][START_REF] Vaughan | Zero-dimensional spaces from linear structures[END_REF] and the references therein to ner properties, the history, and nontrivial examples of ultraparacompact spaces. Denition 8.5. A Hausdor topological space X is ultraparacompact if every open cover U of X admits a renement by clopen pairwise disjoint sets.

Alternatively (see [START_REF] Ellis | Extending continuous functions on zero-dimensional spaces[END_REF]Proposition 1.2]), a Hausdor space X is ultraparacompact if and only if it is paracompact 3 , and if whenever F ⊆ O ⊆ X, where F is closed and O is open, there is a clopen C ⊆ X such that F ⊆ C ⊆ O.

Example 8.6. Recall that a topological space X is Lindelöf if every open cover of X admits a countable subcover. All compact spaces are Lindelöf, and all second-countable spaces are Lindelöf, and there are spaces which are compact but not second-countable and vice-versa.

Let us prove that every Lindelöf, Hausdor and zero-dimensional space X is ultraparacompact. Let U be an open cover of X. Since X is zero-dimensional, there exists a renement V of U by clopen sets, and we may assume that V is countable as X is Lindelöf, say V = {V n : n ∈ N}. Letting V 0 = ∅, and dening (a) There exist a continuous function a : S * X → T such that for every s ∈ S and x ∈ X s * , ϕ(θ s (x)) = γ a(s,x) (ϕ(x)); (b) For every s ∈ S and every x ∈ X s * , there exists a neighbourhood U ⊆ X s * of x and t ∈ T such that ϕ(θ s (x)) = γ t (ϕ(x)) for all x ∈ U .

Proof. Assuming that (a) is valid and given (s, x) ∈ S * X, we take t = a(s, x) and U = {y ∈ X s * : a(s, y) = t}, which is open since T is discrete and a is continuous. Then the statement in (b) is valid. Assume then that (b) is valid. Given s ∈ S, the condition in (b) and ultraparacompactness of X s * allow us to nd a clopen partition U s , and a family {t U : U ∈ U s } ⊆ T such that for all U ∈ U s and all x ∈ U , ϕ(θ s (x)) = γ t U (ϕ(x)).

We dene a : S * X → T , by setting a(s, x) = t U , where U is chosen as the unique element of U s such that x ∈ U . Then (a) holds.

We are now ready to prove that topological isomorphisms between Hausdor groupoids of germs yield a continuous orbit equivalence between the respective partial actions. Theorem 8.9. Suppose that θ and γ are almost ample topological partial actions, and that the groupoids of germs S X and T Y are topologically isomorphic. Then θ and γ are continuously orbit equivalent.

Proof. Let Φ : S X → T Y be an isomorphism of topological groupoids. As (S X) (0) = X and (T Y ) (0) = Y , the restriction ϕ := Φ| X : X → Y is a homeomorphism.

We will use Lemma 8.8. Let s ∈ S and x ∈ X s * be xed. Since [s,

3 A topological space X is paracompact if every open cover of X admits a locally nite renement.

If a ∈ [t, V ], then r(a) = γ t (s(a)). It follows that for all x ∈ U , we have ϕ(θ s ( is constructed in a similar manner, and we therefore obtain a continuous orbit equivalence from θ to γ.

Example 8.10. In some sense, the hypothesis that the domains of the partial actions are ultraparacompact is the weakest condition possible one needs to assume to obtain Theorem 8.9.

For example, suppose that X is Hausdor, but not ultraparacompact (for example, X = ω 1 , the rst uncountable ordinal with the order topology, which is in fact locally compact and zero-dimensional).

Let U be any clopen cover of X which does not admit any renement by pairwise disjoint clopen sets. We let S be the collection of all nite intersections of elements of U , which is a semigroup (actually, a semilattice) under intersection, and let θ = ({X A } A∈S , {θ A } A∈S ) be the natural action of S on X: X A = A and θ A = id A , the identity of A, for all A ∈ S.

Also, let G = {1} be the trivial group and γ the trivial action of G on X: γ 1 = id X . Then both S X and G X are isomorphic, as topological groupoids, to X. Let us prove, however, that θ and γ are not continuously orbit equivalent. Suppose, on the contrary, that (ϕ, a, b) were a continuous orbit equivalence from θ to γ. For all x ∈ X, we have

which in particular implies that

For each A ∈ S, we consider the subset U

Thus V is a renement of S, and therefore a renement of U , contradicting the choice of U . Topological full pseudogroups. We will use a similar terminology to that of [START_REF] Matui | Homology and topological full groups of étale groupoids on totally disconnected spaces[END_REF]. For each compact-open bisection U ∈ G a of an ample groupoid G, we denote by τ U the homeomorphism given by the canonical action of G a on G (0) , namely

Denition 8.12. The topological full pseudogroup of an ample groupoid is the semigroup [START_REF] Orlo | Equivalent groupoids have Morita equivalent Steinberg algebras[END_REF]. If θ is a partial action of an inverse semigroup S on a locally compact Hausdor and zerodimensional space X, then the topological full pseudogroup [[S X]] is the set of all partial homeomorphisms ϕ : U → V of X for which there are s 1 , . . . , s n ∈ S and compact-open U 1 , . . . , U n ⊆ X such that

The proposition below was proven in [START_REF]Cartan subalgebras in C * -algebras[END_REF]Corollary 3.3] when one considers all open bisections instead of only compact-open ones. In any case, we provide a short and direct proof of it. Proposition 8.14. Suppose G is an ample groupoid. Then the homomorphism τ :

Moreover, from id s(V ) = τ V -1 U we also have equality of the domains, s(V ) = s(V -1 U ), which implies

and symmetrically we obtain U ⊆ V . Therefore U = V and τ is injective. Conversely, suppose int(Iso(G)) = G (0) . Take any nonempty compact-open bisection U ⊆ int(Iso(G)) which is not contained in G (0) . Then U = s(U ) but τ U = τ s(U ) , so τ is not injective.

Let us now summarize the connections between continuous orbit equivalence of partial actions, isomorphisms of groupoids of germs, isomorphisms of topological full pseudogroups, diagonal-preserving isomorphisms of Steinberg algebras, and consequently diagonal-preserving isomorphisms of the associated crossed products. To do so, we will use [START_REF]Diagonal-preserving isomorphisms of étale groupoid algebras[END_REF]Corollary 5.8], which is an improvement of [START_REF] Toke | Diagonal-preserving graded isomorphisms of Steinberg algebras[END_REF]Theorem 3.1].

Note that each individual implication in the next theorem is valid under weaker hypotheses (e.g. ( 1) ⇐⇒

(2) does not require that the groupoids of germs are Hausdor ).

Theorem 8.15. Let R be an indecomposable commutative unital ring and suppose that θ and γ are almost ample and topologically principal partial actions, and that the groupoids of germs S X and T Y are Hausdor. Then the following are equivalent:

(1) the partial actions θ and γ are continuously orbit equivalent;

(2) the groupoids of germs S X and T Y are topologically isomorphic;

(3) the inverse semigroups (S X) a and (T Y ) a are topologically isomorphic;

(4) the inverse semigroups [[S X]] and [[T Y ]] are isomorphic;

(5) there exists a diagonal-preserving (ring or R-algebra) isomorphism between the Steinberg algebras A R (S X) and A R (T Y ); (6) there exists a diagonal-preserving (ring or R-algebra) isomorphism between the crossed products A R (X) S and A R (Y ) T .

Proof. ( 1) ⇐⇒ (2) follows from Theorems 8.4 and 8.9.

(2) ⇐⇒ (3) follows from non-commutative Stone duality: See, for example, [START_REF] Lawson | Pseudogroups and their étale groupoids[END_REF]Theorem 3.23]. (Note that Hausdor Boolean groupoids of [START_REF] Lawson | Pseudogroups and their étale groupoids[END_REF] corresponds to ample Hausdor groupoids.)

(3) ⇐⇒ (4) follows from Proposition 8.14.

(2) ⇐⇒ (5) follows from [54, Corollary 5.8.].

(5) ⇐⇒ (6) follows from Theorem 5.10.

Orbit equivalence of graphs and Leavitt path algebras

In [START_REF] Brownlowe | Graph algebras and orbit equivalence[END_REF], the notion of continuous orbit equivalence for directed graphs was introduced, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts (see [START_REF] Matsumoto | Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras[END_REF]). We will compare this notion with the continuous orbit equivalence of canonical actions of inverse semigroups associated to directed graphs.

A similar study was made by Li in [START_REF] Li | Partial transformation groupoids attached to graphs and semigroups[END_REF], who considered the case of partial actions of free groups generated by edges of a graph. We reiterate that we do not make any assumptions on the second-countability of topological spaces, or countability of graphs.

Directed graphs. A directed graph is a tuple E = (E 0 , E 1 , s, r), where E 0 is a set of vertices, E 1 is a set of edges and s, r : E 1 → E 0 are functions, called the source and range.

A path in E is a nite or innite sequence µ = (µ i

Remark 9.1. Even though every groupoid has a structure of graph, the conventions for concatenation do not agree: arrows/edges in groupoids are usually thought of functions, and thus they are read from right to left. On the other hand, the usual convention for paths in a graph is to read them from left to right.

Nevertheless, this shall bring no confusion to our discussion.

The length of a nite path µ is the number |µ| of edges in µ, that is, if µ = µ 1 • • • µ n , where µ i ∈ E 1 , then |µ| = n. Each vertex in E 0 is also regarded as a path of length 0, and each edge in E 1 is a path of length 1. So given an integer n ≥ 0, we denote E n the set of paths of length n. The set of nite paths of E will be denoted by E = ∞ 0≤n<∞ E n . The length of an innite path µ is simply |µ| = ∞, and the set of all innite paths is denoted E ∞ .

The shift map and boundary path groupoid. For each n ∈ N, let ∂E ≥n = {x ∈ ∂E : |x| ≥ n}. Then ∂E ≥n = µ∈E n Z(µ) is an open subset of ∂E. We dene the one-sided shift map σ : ∂E ≥1 → ∂E as follows:

The n-fold composition σ n is dened on ∂E ≥n and we understand σ 0 : ∂E → ∂E as the identity map. Following [START_REF] Brownlowe | Graph algebras and orbit equivalence[END_REF], the boundary path groupoid of E is

where the product and inverse are dened as

and (x, k, y) -1 = (y, -k, x).

As such, G E is a groupoid with unit space G (0) E = {(x, 0, x) : x ∈ ∂E}, which we identify with ∂E. To put a topology on G E , we consider nite paths µ, ν ∈ E with r(µ) = r(ν), and a nite set of edges F ⊆ s -1 (r(µ)).

Then we dene the sets

The collection of these sets provides a basis of compact-open bisections for a Hausdor topology on G E (see [START_REF] Kumjian | Graphs, groupoids, and Cuntz-Krieger algebras[END_REF]Proposition 2.6] for more details in the case of row-nite graphs and [START_REF]Graph inverse semigroups, groupoids and their C * -algebras[END_REF]Section 3] for the general case).

Proposition 9.5. The groupoid of germs S E ∂E, associated to the canonical action of S E on ∂E and the boundary path groupoid G E are isomorphic as topological groupoids.

Proof. The map ψ :

is a surjective semigroupoid homomorphism. Given ((µ i , ν i ), x i ) ∈ S E * ∂E (i = 1, 2), we need to verify the equivalence (9.6)

where ∼ is the germ equivalence relation (Equation (3.2)). This is enough, because it implies that ψ factors (uniquely) to a (semi)groupoid isomorphism between S E ∂E and G E . First, we write x i = ν i x i . Then ψ((µ 1 , ν 1 ), x 1 ) = ψ((µ 2 , ν 2 ), x 2 ) is equivalent to the following three statements (simultaneously):

From items (i) and (iii), it follows that ν 1 and ν 2 are comparable, as are µ 1 and µ 2 . Item (ii) then implies that either both ν 1 and µ 1 are subpaths of ν 2 and µ 2 , respectively, or the reverse is true. By symmetry, let us assume that ν 1 and µ 1 are subpaths of ν 2 and µ 2 , respectively, say ν 2 = ν 1 p and µ 2 = µ 1 q. From item (i) we obtain px 2 = x 1 , and thus from (iii), µ 1 px 2 = µ 1 x 1 = µ 2 x 2 = µ 1 qx 2 , and therefore p = q.

In other words, these three items imply that

since θ E and θ F are actions. (4) Assume that |µ| ≥ 1 and |ν| = 0.

Applying the case 3 to (ν, µ), there exists a neighbourhood

x, as we wanted. (5) Assume that |µ|, |ν| ≥ 1. In this case, (µ, ν) = (µ, r(µ))(r(µ), ν), so we may apply cases 3 and 4, and proceed in a manner similar to that of case 3.

Since we have exhausted all possibilities for (µ, ν), the theorem is proven.

The Leavitt path algebra L R (E) of a directed graph E with coecients in a unital commutative ring R is the R-algebra generated by a set {v ∈ E 0 } of pairwise orthogonal idempotents and a set of variables e, e * : e ∈ E 1 satisfying the relations: (iii) e * f = δ e,f r(e) for all e, f ∈ E 1 (where δ x,y denotes the Kronecker delta); (iv) v = e∈s -1 (v) ee * whenever v is not a sink nor an innite emitter.

The Leavitt path algebra L R (E) is isomorphic to the Steinberg algebra A R (G E ) of the boundary path groupoid G E (see [START_REF] Orlo | Equivalent groupoids have Morita equivalent Steinberg algebras[END_REF]Example 3.2]). By Proposition 9.5 the groupoids G E and S E ∂E are isomorphic, so by Theorem 5.10 we obtain the isomorphisms

Finally, from Propositions 9.12 and Theorem 8.15, we obtain the following theorem: Theorem 9.16. Let E and F be directed graphs that satisfy the Condition (L) and R an indecomposable commutative unital ring. Then the following are equivalent:

(i) the graphs E and F are continuously orbit equivalent;

(ii) the actions θ E and θ F are continuously orbit equivalent;

(iii) S E ∂E and S F ∂F are isomorphic as topological groupoids;

(iv) G E and G F are isomorphic as topological groupoids; (v) there exists a diagonal-preserving isomorphism between the Steinberg algebras A R (G E ) and A R (G F );

(vi) there exists a diagonal-preserving isomorphism between the skew inverse semigroup rings A R (∂E) S E and A R (∂F ) S F ; (vii) there exists a diagonal-preserving isomorphism between the Leavitt path algebras L R (E) and L R (F ).