Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics
Gilles G. de Castro Daniel Gonçalves Benjamin Steinberg



We prove the Effros-Hahn conjecture for groupoid algebras with coefficients in a sheaf, obtaining as a consequence a description of the ideals in skew inverse semigroup rings. We also use the description of the ideals to characterize when the groupoid algebras with coefficients in a sheaf are von Neumann regular, primitive, semiprimitive, or simple. We apply our results to the topological dynamics of actions of inverse semigroups, describing the existence of dense orbits and minimality in terms of primitivity and simplicity, respectively, of the associated algebra. Moreover, we apply our results to the usual complex groupoid algebra of continuous functions with compact support, used to build the C*-algebra associated with a groupoid, and describe criteria for its simplicity.  


Maximal hypersurfaces and aspects of volume of the Kerr family of black holes
    Rahul Nigam0000-0002-0497-5898
    April 29, 2024
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§ INTRODUCTION


Groupoid algebras with coefficients in a sheaf unify the study of usual convolution groupoid algebras (often called Steinberg algebras) and skew inverse semigroup rings (see <cit.>). The aforementioned constructions are key in the study of algebras associated with combinatorial objects such as graphs, higher rank graphs, ultragraphs, etc, and have deep connections with topological dynamics and the intrinsic dynamics associated with combinatorial objects, see <cit.> for example. Among the relevant properties of convolution groupoid algebras and inverse semigroup skew rings, the constitution of its ideals play a crucial role. For just a couple of examples, the ideal structure of a Leavitt path algebra may be recovered from the ideal structure of a convolution groupoid algebra, see <cit.>, and minimality and topological freeness of actions of inverse semigroups can be described in terms of simplicity of the associated skew rings (see <cit.>). Our goal in this paper is to obtain a description of the ideals in groupoid algebras with coefficients in a sheaf and apply this description to the topological dynamics of actions of inverse semigroups and to the usual complex groupoid algebra of continuous functions with compact support used to build the C*-algebra associated with a groupoid. 

When one searches the literature for a classification of ideals in crossed products, the references go back to the original Effros-Hahn conjecture, which suggested that every primitive ideal of a crossed product of an amenable locally compact group with a commutative C*-algebra should be induced from a primitive ideal of an isotropy group, see <cit.>. Since then, the conjecture has been proved and generalized in several contexts, see <cit.> for a comprehensive account of developments. 

In the purely algebraic setting, an Effros-Hahn type conjecture is proved for partial skew group rings in <cit.> and for groupoid convolution algebras in <cit.> and <cit.>. In this paper, we prove an algebraic Effros-Hahn type conjecture in the context of groupoid algebras with coefficients in a sheaf. Therefore, we extend the known results of <cit.>  to include skew inverse semigroup rings, and at the same time provide a unified statement to the Effros-Hahn conjecture proved in <cit.>. 

After we show our version of the Effros-Hanh conjecture, we use the machinery developed to prove it, namely the induction of modules for groupoid algebras with coefficients in a sheaf,  to describe several algebraic properties of groupoid algebras with coefficients in a sheaf. This includes simplicity,  semiprimitivity, and primitivity. For topological actions of inverse semigroups, since the associated skew inverse semigroup ring can be seen as a groupoid convolution algebra with coefficients in a sheaf, we relate topological properties of the action with algebraic properties of the associated groupoid convolution algebra. Furthermore, we realize the usual algebra C_c(𝒢)  (used to build groupoid C^*-algebras) as a groupoid algebra with coefficients in a sheaf (and hence as a skew inverse semigroup ring), and then apply our topological results to describe when it is simple in terms of the groupoid. Therefore, we provide a bridge between algebra and analysis (for example, our simplicity characterization of C_c(𝒢) should be compared with the characterization of simplicity of the reduced groupoid C*-algebra).



We now give a more detailed description of our work.

We begin by presenting some necessary background on the topics of the paper. In particular, we recall the construction of convolution algebras with coefficients in a sheaf and of skew inverse semigroup rings. Furthermore, we recall the Disintegration theorem, which is used in <cit.> to prove that skew inverse semigroup rings and groupoid convolution algebras with coefficients in a sheaf are essentially the same objects. 

We describe the induction process that transforms representations of the skew group rings associated with isotropy groups to modules for groupoid algebras with coefficients in a sheaf in Section <ref>, and use this to prove Theorem <ref>, which asserts that every ideal in a groupoid algebra with coefficients in a sheaf of modules is an intersection of annihilators of induced modules.

In section <ref>, we prove the Effros-Hahn conjecture for 
groupoid convolution algebras with coefficients in a sheaf, Theorem <ref>,
and use Theorem <ref> to prove a number of properties of , the groupoid convolution algebra associated with a sheaf. We start by showing that every primitive ideal of  is the annihilator of a single induced representation, Theorem <ref>. We then show that induction of modules preserves simplicity, Theorem <ref>. As with usual groupoid algebras, there is a diagonal commutative algebra inside , call it . We characterize when  is von Neumann regular in Proposition <ref>, and describe when it is maximal commutative inside  in Proposition <ref>. We specify the role of  in determining when a ring homomorphism from  is injective in the Generalized Uniqueness Theorem, Theorem <ref>. We describe primitivity of  in Theorem <ref> and, under some assumptions, show that  is left primitive if, and only if, the unit space of the groupoid has a dense orbit, Theorem <ref>. In Theorems <ref> and <ref> we give a sufficient condition for  to be semiprimitive and in Theorem <ref> we characterize simplicity of .

We devote Section <ref> to the topological dynamics of actions of inverse semigroups. In Proposition <ref>, we show that if the associated algebra Γ_c(S⋉ X,) is left primitive then the action has a dense orbit. From this, we obtain that for certain topologically free actions on locally compact, Hausdorff, zero-dimensional spaces the associated algebra Γ_c(S⋉ X,) is left primitive if, and only if, the action has a dense orbit, Corollary <ref>. In Propositions <ref> and <ref> we give sufficient conditions for semiprimitivity of Γ_c(S⋉ X,), and, in Theorem <ref> we relate simplicity of Γ_c(S⋉ X,) with minimality of the action. 

Finally, in Section <ref>, we study the usual algebra C_c() of complex valued, continuous functions with compact support, which is used to build the C*-algebras (full and reduced) associated to a groupoid. For Hausdorff groupoids, we give a direct description of C_c() as a skew inverse semigroup ring, Proposition <ref>. For general groupoids, we provide a realization of C_c() as a groupoid convolution algebra with coefficient in a sheaf in Theorem <ref>; this implies that the same skew inverse semigroup ring representation as in the Hausdorff case holds in general. Applying results of the previous section to C_c(), in the Hausdorff case, we characterize the simplicity of C_c() in terms of minimality and effectiveness of , see Theorem <ref>. We finish the paper considering the groupoid arising from a partial action and prove that C_c() is a partial crossed product, where  is the transformation groupoid of the partial action. 








§ BACKGROUND


For completeness, in this section we recall the relevant concepts that will be used throughout the paper, as defined in <cit.>.



 §.§ Groupoids


A groupoid 𝒢 is a small category of isomorphisms. A topological groupoid is a groupoid equipped with a topology making the multiplication and inverse operations continuous. The elements of the form gg^-1 are called units. We denote the set of units of  𝒢 by 𝒢^(0), and refer to 𝒢^(0) as the unit space.  The source and range maps are given by (g)=g^-1g and (g)= gg^-1, for g∈ G. These maps are necessarily continuous when G is a topological groupoid.

An étale groupoid is a topological groupoid  𝒢 such that its unit space  𝒢 0 is locally compact and Hausdorff and its range map  is a local homeomorphism (this implies that the domain map  and the multiplication map are also local homeomorphisms). A bisection of 𝒢 is a subset B⊆𝒢 such that the restriction of the range and source maps to B are injective. An étale groupoid is ample if its unit space has a basis of compact open sets or, equivalently, if the arrow space 𝒢 1 has a basis of compact open bisections (when it is clear from the context, we will also use  to denote the arrow space). We remark for future use that, for every open bisection B, of an ample groupoid 𝒢, the range and source maps are homeomorphisms from B to (B) and (B), respectively. The isotropy group of a unit x∈𝒢^(0) is the group 𝒢_x^x = {g∈𝒢|(g)=(g)=x}. The isotropy subgroupoid of a groupoid 𝒢 is
the subgroupoid 
    Iso(𝒢)=⋃_x∈𝒢^(0)𝒢_x^x ={g∈𝒢: (g)=(g)}.
 We say that the topological groupoid 𝒢 is effective if 𝒢^(0)= int(Iso(𝒢)).
 We will use the notation ζ:y→ z meaning that (ζ)=y and (ζ)=z. Furthermore, for each x ∈𝒢 0, Orb(x) = ∘^-1(x). 
 



From now on, following Bourbaki, the term “compact” will include the Hausdorff axiom.  However, a space can be locally compact without being Hausdorff.
If f X→ Z and g Y→ Z are maps of spaces, then their pullback is 
    X×_f,g Y={(x,y)| f(x)=g(y)}
 (with the subspace topology of the product space).





 §.§ Ample groupoid convolution algebras with coefficients in a sheaf of rings

In this section, we recall the convolution algebra associated with a sheaf of rings over an ample groupoid[Technically, this is just a ring but convolution algebra fits better with terminology in operator algebras and every ring is a ℤ-algebra]. 




Let 𝒢 be an ample groupoid.  Then a 𝒢-sheaf ℰ consists of a topological space E, a local homeomorphism p E→𝒢 0 and a continuous map α𝒢 1×_,p E→ E (written (γ,e)↦α_γ(e)) satisfying the following axioms:


  (S1) α_p(e)(e) = e;

  (S2) p(α_γ(e))=(γ) if (γ)=p(e);

  (S3) α_β(α_γ(e)) = α_βγ(e) whenever (β)=(γ) and (γ)=p(e).
								
If x∈𝒢 0, then ℰ_x=p^-1(x) is called the stalk of ℰ at x.  Notice that α_γℰ_(γ)→ℰ_(γ) is a bijection with inverse α_γ.   









We shall be interested in sheaves with extra structure.  A 𝒢-sheaf of (unital) rings is a 𝒢-sheaf 𝒪=(E,p,α) equipped with a unital ring structure on each stalk 𝒪_x such that the following axioms hold:


  (SR1) + E×_p,p E→ E is continuous;

  (SR2) · E×_p,p E→ E is continuous;

  (SR3) the unit section x↦ 1_x is a continuous mapping 𝒢 0→ E;

  (SR4) α_γ𝒪_(γ)→𝒪_(γ) is a ring homomorphism for all γ∈𝒢 1.


Note that the zero section x↦ 0_x is continuous and that the negation map is continuous (these are standard facts about sheaves of abelian groups, and hence rings, over spaces, cf. <cit.>).

Given a 𝒢-sheaf of rings 𝒪=(E,p,α,+,·), next we recall the definition of the ring of global sections of 𝒪 with compact support, which we shall also call the convolution algebra of 𝒢 with coefficients in the sheaf of rings 𝒪.
Let A(𝒢,𝒪) be the set of all mappings f𝒢 1→ E such that p∘ f=, that is, f(γ)∈𝒪_(γ) for all γ∈𝒢 1. Equip A(𝒢,𝒪) with a binary operation by defining (f+g)(γ)= f(γ)+g(γ), which we refer to as pointwise addition. With this operation, A(𝒢,𝒪) is an abelian group with respect to pointwise addition with 0 as the identity and (-f)(γ) = -f(γ) for γ∈𝒢 1.

We define, as an abelian group, Γ_c(𝒢,𝒪) to be the subgroup generated by all mappings f∈ A(𝒢,𝒪) such that there is a compact open bisection U with f|_U continuous and f|_𝒢 1∖ U = 0. In this case, we say that f is supported on U. If U is a compact open bisection and s(U)→ E is any (continuous) section of p, then we can define an element sχ_U∈Γ_c(𝒢,𝒪), supported on U, by

    (sχ_U)(γ) =  s((γ)),    if γ∈ U
     0_(γ),    else.
  In the special case that s is the unit section x↦ 1_x over U, we denote sχ_U by simply χ_U.  In other words,

    χ_U(γ) =  1_(γ),    if γ∈ U
     0_(γ),    else.


Notice that if f∈Γ_c(𝒢,𝒪) is supported on a compact open bisection U, then f=sχ_U where s=f∘ (|_U).  Thus Γ_c(𝒢,𝒪) can also be described as the abelian group generated by all elements of the form sχ_U where s(U)→ E is a section, and U is a compact open bisection.



A crucial property of elements of Γ_c(𝒢,𝒪), and that we will use in our work, is that they can only be non-zero on finitely many points of any fiber of  or .


Let f∈Γ_c(𝒢,𝒪) and x∈𝒢 0.  Then there are only finitely many γ∈ (x) such that f(γ)≠ 0 and, similarly, for (x).



Finally, to make Γ_c(𝒢,𝒪) into a ring, we define the convolution of elements of Γ_c(𝒢,𝒪) as follows. 
If f,g∈Γ_c(𝒢,𝒪) and γ∈𝒢 1, then

    f∗ g(γ) = ∑_βρ=γf(β) α_β(g(ρ)).



It is proved in <cit.> that 
if f,g∈Γ_c(𝒢,𝒪), then f∗ g∈Γ_c(𝒢,𝒪). In fact it is shown that if f is supported on U and g is supported on V, with U,V compact open bisections, then f∗ g is supported on UV.  In the case that  is Hausdorff, it is shown in <cit.> that   consists precisely of those continuous functions f^(1)→ E with p∘ f= and compact support (i.e., the inverse image of the complement of the zero section is compact).   Furthermore, in <cit.> it is shown how to build a sheaf of rings so that the above construction yields the usual algebra of 𝒢 over a unital ring R from <cit.>. This sheaf is recalled below.



 Let R be any unital ring, which we view as a space with the discrete topology.  We define the constant sheaf of rings Δ(R) to be the 𝒢-sheaf of rings with E=R×𝒢 0 and with p R×𝒢 0→𝒢 0 the projection.  The addition and multiplication are pointwise, that is, (r,x)+(r',x) = (r+r',x) and (r,x)(r',x) = (rr',x).  The mapping α is given by α(γ)(r,(γ)) = (r,(γ)).  Then Γ_c(𝒢,Δ(R)) is the usual algebra of 𝒢 over R, from <cit.>.







 §.§ The disintegration theorem



A key result in <cit.> is the disintegration theorem, which generalizes results in <cit.> for Steinberg algebras. In this subsection, we recall the theorem as well as the related concepts.

Let 𝒢 be an ample groupoid and 𝒪=(E,p,α) be a 𝒢-sheaf of rings.  Put R=Γ_c(𝒢,𝒪); it is a ring with local units.  A (left) R-module M is unitary if RM=M.  We denote by R the category of unitary (left) R-modules. 

A 𝒢-sheaf of 𝒪-modules ℳ=(F,q,β) is a 𝒢-sheaf such that each stalk ℳ_x has a (unitary) left 𝒪_x-module structure such that:

  
  (SM1) addition + F×_q,q F→ F is continuous;
  
  (SM2) the  module action E×_p,q F→ F is continuous;
  
  (SM3) β_γ(rm) = α_γ(r)β_γ(m) for all r∈𝒪_(γ) and m∈ℳ_(γ).



The following result is proved in <cit.> and will be used in our text.


Let X be a Hausdorff space with a basis of compact open sets and let 𝒜=(F,q) be a sheaf of abelian groups on X. Then, for each f∈𝒜_x, there is a section s X→ F with compact support such that s(x)=f.



If ℳ is a 𝒢-sheaf of 𝒪-modules, then we can look at the set M=Γ_c(𝒢,ℳ) of continuous sections s𝒢 0→ F with compact support. This is an abelian group with pointwise operations.  We define an R-module structure on it by putting, for f∈ R and m∈ M,

    (fm)(x) = ∑_γ∈(x)f(γ)β_γ(m((γ))).




The construction ℳ⟼Γ_c(𝒢,ℳ) is a functor from the category of 𝒢-sheaves of 𝒪-modules to the category of Γ_c(𝒢,𝒪).


A quasi-inverse for the functor above is constructed in <cit.>. We recall the main concepts below. 


Let M be a unitary R-module and x∈𝒢 0. Let 
    N_x = {m∈ M|χ_Um=0 for some U⊆𝒢 0 compact open with x∈ U},
 and define
M_x:= M/N_x. Furthermore, define 
    F= ∐_x∈𝒢 0 M_x
 and q F→𝒢 0 by q([m]_x) = x.  Put a topology on F by taking as a basis all sets of the form

    D(m,U) = {[m]_x| x∈ U},
 where m∈ M and U⊆𝒢 0 is compact open.  To define the 𝒢-sheaf structure, for γ∈𝒢, put 
    β_γ([m]_(γ)) = [χ_Um]_(γ),
 where U is any compact open bisection containing γ.  Then β𝒢 1×_,q F→ F is well defined, continuous, and turns 
    Sh(M) := (F,q,β)
 into a 𝒢-sheaf.  
Moreover, Sh(M) is a  𝒢-sheaf of abelian groups with respect to the fiberwise addition [m]_x+[n]_x=[m+n]_x (i.e., addition is fiberwise continuous and each β_γ is an additive homomorphism). To define the 𝒪_x-module structure on M_x, let r∈𝒪_x. Choose a section t∈Γ_c(𝒢 0,𝒪) with t(x) = r (using Proposition <ref>), and define r[m]_x = [tm]_x. Then Sh(M) is a 𝒢-sheaf of 𝒪-modules and, as proved in <cit.>, we have that the construction  M⟼Sh(M) is a functor, as we make precise below.






The construction M⟼Sh(M) is a functor from the category Γ_c(𝒢,𝒪) to the category of 𝒢-sheaves of 𝒪-modules.


Combining the last two propositions we get.

[Disintegration theorem]
The functors M⟼Sh(M) and ℳ⟼Γ_c(𝒢,𝒪) provide an equivalence between the category Γ_c(𝒢,𝒪) of unitary Γ_c(𝒢,𝒪)-modules and the category of 𝒢-sheaves of 𝒪-modules.



It follows from the above that there are natural isomorphisms M≅Γ_c(𝒢,Sh(M)) and ℳ≅Sh(Γ_c(𝒢,ℳ)).  



As a consequence of Theorem <ref>, M≅Γ_c(𝒢,Sh(M)) for any unitary module M and so we have an equality of annihilator ideals 
    (M)=(Γ_c(𝒢,Sh(M))).
 In particular, for every ideal I of  we have 
    I=( /I)=(Γ_c(Sh(/I))).
 





 §.§ Skew inverse semigroup rings




By a partial automorphism of a ring A, we mean a ring isomorphism φ I→ J between two-sided ideals I,J of A.  The collection of all partial automorphisms of A forms an inverse monoid that we denote I_A.  If S is an inverse semigroup, then an action of S on A is a homomorphism α S→ I_A, usually written s↦α_s.  The domain of α(s) is denoted D_s^* and the range is then D_s.  We say that the action is non-degenerate if 
    ∑_e∈ E(S) D_e=A.


 To ensure associativity of the skew inverse semigroup ring, we assume that each D_s is a ring with local units (although weaker conditions suffice). Given an action α of S on a ring A,
the construction of the corresponding skew inverse semigroup ring is done in three steps.

	
  * First we consider the set

    ℒ = {∑_s∈ S^finite a_s δ_s | a_s ∈ D_s }≅⊕_s∈ S D_s

where δ_s, for s∈ S, is a formal symbol (and 0δ_s=0).
We equip ℒ with component-wise addition and with multiplication
defined as the linear extension of the rule

    (a_s δ_s)(b_t δ_t) = α_s(α_s^*(a_s) b_t) δ_st.


	
  * Then, we consider the ideal
	
    𝒩 = ⟨ a δ_r - a δ_s | r,s ∈ S,  r ≤ s  and  a ∈ D_r ⟩,

i.e., 𝒩 is the ideal of ℒ generated by all elements of the form aδ_r - aδ_s, where r≤ s and a ∈ D_r.   It is shown in <cit.>, that these elements already generate 𝒩 as an additive group.
	
  * Finally, we define the corresponding
	skew inverse semigroup ring, which we denote by A⋊ S, as the quotient ring ℒ/𝒩.

If S is a group, then the ideal 𝒩 is the zero ideal and the multiplication simplifies to the rule aδ_s· bδ_t = aα_s(b)δ_st, and so A⋊ S is the familiar skew group ring. Notice also that the quotient of ℒ by any ideal yields a system, in the sense of <cit.>.


A key class of inverse semigroup actions is that of spectral actions. Recall from <cit.> that an action α of an inverse semigroup S on A is called spectral if it is non-degenerate and D_e has a unit element 1_e for each e∈ E(S), which is necessarily a central idempotent of A. In the case of a spectral action, the central idempotents in the Boolean algebra generated by the 1_e are a set of local units for A.




 §.§ The interplay between groupoid convolution algebras with coefficients in a sheaf of rings and skew inverse semigroup rings.


It is shown in <cit.> that, under certain mild conditions, groupoid convolution algebras with coefficients in a sheaf of rings can be realized as skew inverse semigroup rings and vice versa. We recall this more precisely below.

We denote by 𝒢^a the set of all compact open bisections of 𝒢. It is an inverse semigroup with operations given by 
    BC = { bc ∈𝒢| b∈ B, c∈ C,  and (b) = (c)}
 and B^*={b^-1| b∈ B}. 

Let 𝒪=(E,p,α) be a 𝒢-sheaf of rings.  Put A=Γ_c(𝒢 0,𝒪), which we view as the ring of compactly supported continuous sections of 𝒪 over 𝒢 0 with pointwise operations. In <cit.> a spectral action of S=^a on A is defined. 
If U⊆𝒢 0 is open then we put 𝒪|_U = (p(U),+,·); it is a sheaf of rings on U.  Then A(U)=Γ_c(U,𝒪|_U) can be identified with the subring of A consisting of sections supported on U. For s∈ S, put D_s = A((s)).  Note that since (s) is compact open, D_s has an identity, the mapping χ_(s).   We define an isomorphism α_s D_s^*→ D_s by

    α_s(f)((γ)) = α_γ(f((γ)))
 for γ∈ s and f∈ D_s^*. We then have the following (which is proved more generally for inverse subsemigroups of ^a satisfying certain conditions).

[<cit.>]
Let 𝒪 be a 𝒢-sheaf of rings on an ample groupoid 𝒢. Then, 
    Γ_c(𝒢,𝒪)≅Γ_c(𝒢 0,𝒪)⋊𝒢^a

as rings.


To obtain the converse characterization, we recall some key concepts first.

Given a Boolean action ρ of an inverse semigroup S on a space X, the groupoid of germs 𝒢=S⋉ X is defined as follows (see <cit.>).  The unit space 𝒢 0 is taken to be X and 
    𝒢 1 = {(s,x)∈ S× X| x∈ D_s^*}/∼
 where (s,x)∼ (t,y) if and only if x=y and there exists u≤ s,t with x∈ D_u^*.  We write [s,x] for the class of (s,x). The source and range maps are defined by ([s,x]) = x and ([s,x]) = ρ_s(x), the product is defined by [s,ρ_t(x)][t,x] = [st,x], and the inverse is given by [s,x] = [s^*,ρ_s(x)].  The topology on 𝒢 0 is that of X, whereas a basis of neighborhoods for 𝒢 1 is given by the sets (s,U), where U⊆ D_s^* is open (compact open if X is zero-dimensional), and (s,U)={[s,x]| x∈ U}.

We also need to recall the generalized Stone space of a generalized Boolean algebra B.  A character of B is a non-zero Boolean algebra homomorphism λ B→{0,1} to the two-element Boolean algebra.  
The (generalized) Stone space of B is the space B of characters of B topologized by taking as a basis the sets

    D(a)={λ∈B|λ(a)=1}
 with a∈ B. 


Let α be a spectral action of an inverse semigroup S on a ring A.
We want to define a Boolean action of S on the Pierce spectrum A of A (recall from <cit.> that the Pierce spectrum of A is B, where 
B=E(Z(A))). 

For s∈ S, let 
    D_s = {λ∈ A|λ(1_ss^*)=1}.
   Notice that D_s is compact open and can be identified with the Pierce spectrum of D_s.   Define α_s D_s^*→ D_s by 
    α_s(λ)(e) = λ (α_s^*(e1_ss^*))
 for e∈ E(Z(A)).

Let 𝒢=S⋉ A be the corresponding groupoid of germs.  It is ample as A has a basis of compact open sets. We want to define a 𝒢-sheaf of 𝒪_A rings such that A⋊ S≅Γ_c(𝒢,𝒪_A). For this, 
𝒪_A is the sheaf of unital rings constructed as follows:
The underlying space of 𝒪_A is defined to be E=∐_λ∈E(Z(A)) A/I_λ, where 
    I_λ = {a∈ A|∃ e∈λ(1) with ea=0}
 and the class a+I_λ is denoted [a]_λ. If r∈ A and e∈ E(Z(A)), then we put

    (r,D(e)) = {[r]_λ|λ∈ D(e)}.
   The sets of the form (r,D(e)) form a basis for a topology on E and p E→B, defined by p([r]_λ) = λ, maps (r,D(e)) homeomorphically to D(e), whence p is a local homeomorphism.  The ring structures on the A/I_λ turn 𝒪_A=(E,p,+,·) into a sheaf of unital rings on E(Z(A)). Each stalk of O_A is 𝒪_A,λ = A/I_λ, and the unit is the class [e]_λ where λ(e)=1.  We obtain the 𝒢-sheaf structure on 𝒪_A by putting

    α_[s,λ]([a]_λ) = [α_s(1_s^*sa)]_α_s(λ)

for [s,λ]∈𝒢 1.  

[<cit.>]
If S is an inverse semigroup with a spectral action α on a ring A, then A⋊ S≅Γ_c(S⋉ A,𝒪_A) with the S⋉ A-sheaf  structure on 𝒪_A coming from (<ref>) and the usual sheaf of rings structure on 𝒪_A over the Pierce spectrum A.










§ INDUCED MODULES


In this section, we describe the induction of modules from isotropy skew group rings for groupoid algebras with coefficients in a sheaf, and we show how we can use induction to study the ideals of this algebra. We fix an ample groupoid  and a -sheaf of rings . 

Given x∈^(0), we define L_x={γ∈|(γ)=x} and _x=⊕_γ∈ L_x_(γ) as a direct sum of abelian groups. For γ∈ L_x, 1_γ∈_x represents the element that is 1_(γ) at coordinate γ and it is 0_(η) at all other coordinates η∈ L_x with η≠γ. Since the isotropy group ^x_x of x acts on _x, we can form the skew group ring 
    B_x:=_x⋊^x_x
 which we call the isotropy skew group ring at x.


The abelian group _x has a structure of free right B_x-module where the right action is given by

    a1_γ· bδ = aα_γ(b)1_γδ

for γ∈ L_x, a∈_(γ), b∈_x and δ∈^x_x, and extended in the natural way. Moreover, if for each y∈Orb(x), we choose η_y∈ such that (η_y)=x and (η_y)=y, then {1_η_y}_y∈Orb(x) is a basis for _x.



We check the associativity of the right action, the other properties being straightforward. For that, consider γ∈ L_x, a∈_(γ), b,c∈_x and δ,η∈^x_x. We have that

    (a1_γ· bδ)· cη=aα_γ(b)1_γδ· cη=aα_γ(b)α_γδ(c)1_γδη=


    aα_γ(bα_δ(c))1_γδη=a1_γ· bα_δ(c)δη=a1_γ·(bδ cη).



To prove the second part of the statement, for each y∈Orb(x), we fix an element η_y∈ such that (η_y)=x and (η_y)=y. In order to show that _x is free we consider the abelian group homomorphism Φ:_x→⊕_y∈Orb(x)B_x1_y given by

    Φ(a1_γ)=α_η_(γ)(a)η_(γ)γ1_(γ),

where 1_y is the canonical vector for y∈Orb(x), and ⊕_y∈Orb(x)B_x1_y has the natural right B_x-module structure. Observe that α_η_(γ)(a)∈_(η_(γ))=_x and (η_(γ)γ)=(η_(γ)γ)=x, so that Φ is indeed well-defined. We now show that Φ is a right B_x-module isomorphism.

Given γ∈ L_x, a∈_(γ), b∈_x and δ∈^x_x, we have that

    Φ(a1_γ· bδ)=Φ(aα_γ(b)1_γδ)=α_η_(γδ)(aα_γ(b))η_(γδ)γδ1_(γδ)=


    α_η_(γ)(a)α_η_(γ)γ(b)η_(γ)γδ1_(γ)=α_η_(γ)(a)η_(γ)γ bδ1_(γ)=


    (α_η_(γ)(a)η_(γ)γ 1_(γ))bδ=Φ(a1_γ)bδ,

so that Φ is a right B_x-module homomorphism.


In order to build the inverse of Φ we use the following identification as abelian groups, ⊕_y∈Orb(x)B_x1_y=⊕_y∈Orb(x)⊕_δ∈^x_x_xδ1_y. We now define the abelian group homomorphism Ψ:⊕_y∈Orb(x)B_x1_y→_x by

    Ψ(bδ1_y)=α_η_y(b)1_η_yδ,

where b∈_x, δ∈^x_x and y∈Orb(x). Here, η_yδ∈ L_x and α_η_y(b)∈(η_y)=(η_yδ) so that Ψ is well-defined.

We now prove that Ψ=Φ. Given γ∈ L_x and a∈_(γ), we have that

    Ψ(Φ(a1_γ))=Ψ(α_η_(γ)(a)η_(γ)γ1_(γ))=α_η_(γ)(α_η_(γ)(a))1_η_(γ)η_(γ)γ=a1_γ.

On the other hand, given b∈_x, δ∈^x_x and y∈Orb(x), we have that

    [                                                 Φ(Ψ(bδ1_y)) =Φ(α_η_y(b)1_η_yδ)=α_η_(η_yδ)(α_η_y(b))η_(η_yδ)η_yδ1_(η_yδ);                                                            ;                                                                                   = α^-1_η_y(α_η_y(b))η_yη_yδ1_y=bδ1_y. ]


It follows that Φ is a right B_x-module isomorphism between _x and the free module ⊕_y∈Orb(x)B_x1_y. Moreover, for each y∈Orb(x), Ψ(1_y)=1_η_y so that {1_η_y}_y∈Orb(x) is a basis for _x.



Let x∈^(0). From now on we assume that, for each y∈Orb(x), we have chosen η_y∈ such that η_x=x and, for y≠ x, 
    (η_y)=x  and (η_y)=y.



The abelian group _x has a structure of Γ_c(𝒢,𝒪)-left module with left action given by

    f· a1_γ=∑_(β)=(γ)f(β)α_β(a)1_βγ,

where f∈Γ_c(𝒢,𝒪), γ∈ L_x and a∈_(γ). Moreover, with the B_x right action given by Equation (<ref>), we have that _x is a Γ_c(𝒢,𝒪)-B_x-bimodule.



By Proposition <ref>, the sum in (<ref>) is finite. Also notice that f(β),α_β(a)∈_(β)=_(βγ), so that the expression in (<ref>) is well-defined. For the left-module properties, we prove the associativity, the other properties being immediate. For f,g∈Γ_c(𝒢,𝒪), γ∈ L_x and a∈_(γ), we have that

    f·(g· a1_γ)     =f·(∑_(β)=(γ)g(β)α_β(a)1_βγ) 
        =∑_(β)=(γ)∑_(δ)=(β)f(δ)α_δ(g(β)α_β(a))1_δβγ
        =∑_(β)=(γ)∑_(δ)=(β)f(δ)α_δ(g(β))α_δβ(a)1_δβγ
        =∑_(ζ)=(γ)∑_δβ=ζf(δ)α_δ(g(β))α_δβ(a)1_δβγ
        =∑_(ζ)=(γ)(f*g)(ζ)α_ζ(a)1_ζγ
       =(f*g)· a1_γ.


We now check the associativity with respect to the left and right actions. Fix f∈Γ_c(𝒢,𝒪), γ∈ L_x, a∈_(γ), b∈_x and δ∈^x_x. Then,

    (f· a1_γ)· bδ    = (∑_(β)=(γ)f(β)α_β(a)1_βγ)· bδ
       =∑_(β)=(γ)f(β)α_β(a)α_βγ(b)1_βγδ
       =∑_(β)=(γ)f(β)α_β(aα_γ(b))1_βγδ
       =f·(aα_γ(b)1_γδ)
       =f·(a1γ· bδ).




A sum such as in (<ref>) can be decomposed using orbits. In general, for x∈^(0) the sum ∑_(β)=x can be decomposed as ∑_y∈Orb(x)∑_γ:x→ y, where the last sum is over all γ∈ such that (γ)=x and (γ)=y.


In possession of the bimodule _x, we can define an induction functor  _x:B_x→Γ_c(𝒢,𝒪) defined by 
    _x(M)=_x⊗_B_x M.
 The induction functor is exact as _x is a free right B_x-module. We can also use the basis of Proposition <ref> to decompose _x(M) as a direct sum. More specifically, for each y∈Orb(x) let η_y∈ as in (<ref>). Then, given M∈B_x, we have that

    _x(M)=_x⊗_B_x M=⊕_y∈Orb(x)1_η_y⊗ M.

We now compute how the Γ_c(𝒢,𝒪) action behaves with respect to the isomorphism decomposition of _x(M). Given f∈Γ_c(𝒢,𝒪), y∈Orb(x) and m∈ M, we have that

    f· 1_η_y⊗ m     = ∑_z∈Orb(x)∑_ζ:y→ zf(ζ)1_ζη_y⊗ m 
       =∑_z∈Orb(x)∑_ζ:y→ zα_η_zη_z(f(ζ))1_η_zη_zζη_y⊗ m 
       =∑_z∈Orb(x)∑_ζ:y→ z1_η_z⊗α_η_z(f(ζ))η_zζη_y · m,

where the first equality follows from (<ref>) and the decomposition of the set {β∈|(β)=y} as ⋃_z∈Orb(x){ζ∈|(ζ)=y,(ζ)=z}.

 

The following lemma follows immediately from Equation (<ref>).


Let M be a B_x, _x(M) be the induced module, and for each y∈Orb(x) let η_y∈ as in (<ref>). Then, for every f∈Γ_c(𝒢,𝒪), we have that f∈(_x(M)) if, and only if, for every y,z∈Orb(x), we have that ∑_ζ:y→ zα_η_z(f(ζ))η_zζη_y∈(M).


Now let ℳ be a 𝒢-sheaf of 𝒪-modules. Our next goal is to describe (Γ_c(𝒢,ℳ)) in terms of the annihilators of the induced modules. First, we have to give a structure of left B_x-module to ℳ_x for each x∈^(0). Given b∈_x, δ∈^x_x and m∈ℳ_x, we define

    bδ· m:=b β_δ(m).

As before, we only prove the associativity of the left product. Suppose we are given as well, c∈_x and η∈^x_x, then

    bδ· (cη· m)=bδ· (cβ_η(m))=bβ_δ(cβ_η(m))=bα_δ(c)β_δη(m)=


    (bα_δ(c)δη)· m=(bδ cη)· m.


The following generalizes <cit.>. We adapt the proof there to accommodate the extra structure.


Let ℳ be a 𝒢-sheaf of 𝒪-modules. Then,

    (Γ_c(𝒢,ℳ))=⋂_x∈^(0)(_x(ℳ_x)).

Consequently, every ideal of Γ_c(𝒢,𝒪) is an intersection of annihilators of induced modules.



Let f∈(Γ_c(𝒢,ℳ)) and x∈^(0). In order to prove that f∈(_x(ℳ_x)), by Lemma <ref>, it suffices to show that for every y,z∈Orb(x), we have that ∑_ζ:y→ zα_η_z(f(ζ))η_zζη_y∈(ℳ_x). Let m∈ℳ_x. Then, by Proposition <ref>, there exists s∈Γ_c(𝒢,ℳ) such that s(x)=m. Fix y,z∈Orb(x). By Proposition <ref>, |(z)∩(f)|<∞, and since ^(0) is Hausdorff, there exists a compact-open neighborhood U of y such that U∩((z)∩(f)){y}. Now, let U_y and U_z be compact-open bisections such that η_y∈ U_y and η_z∈ U_z. By replacing U_y with UU_y, if necessary, we may assume that (U_y) U. Since f∈(Γ_c(𝒢,ℳ)) and the annihilator is an ideal, we have that χ_U_z*f*χ_U_y∈(Γ_c(𝒢,ℳ)), and therefore

    0=((χ_U_z*f*χ_U_y)s)(x)=∑_γ∈(x)(χ_U_z*f*χ_U_y)(γ)β_γ(s((γ))).

Notice that, for each γ∈(x)

    (χ_U_z*f*χ_U_y)(γ)=∑_γ_1γ_2γ_3=γχ_U_z(γ_1)α_γ_1(f(γ_2))α_γ_1γ_2(χ_U_y(γ_3)).

We are interested in the non-zero terms of the above sum. For that, it is necessary that γ_1=η_z, since η_z is the sole element of U_z such that the range is x. Also, by the properties of U and U_y, it is necessary that (γ_2)=y and hence γ_3=η_y. It follows that

    (χ_U_z*f*χ_U_y)(γ)=∑_ζ:y→ z 
    γ=η_zζη_yα_η_z(f(ζ)).

By substituting (<ref>) in (<ref>) and making the appropriate change of variables, we get

    0   =∑_ζ:y→ zα_η_z(f(ζ))β_η_zζη_y(s(x)) 
       =∑_ζ:y→ zα_η_z(f(ζ))η_zζη_y· m.

Since m was arbitrary, ∑_ζ:y→ zα_η_z(f(ζ))η_zζη_y∈(ℳ_x) as we wanted to prove.

Suppose now that f∈(_x(ℳ_x)) for every x∈^(0). Given s∈, we want to prove that fs=0. For x∈^(0), by Lemma <ref>, if y∈Orb(x)  and η_y∈ is as in (<ref>) then, we have that

    0     =∑_γ:x→ yα_η_y(f(γ))η_yγ· s(x) 
       =∑_γ:x→ yα_η_y(f(γ))β_η_yγ(s(x)) 
       =β_η_y(∑_γ:x→ y f(γ)β_γ(s(x))),

and since β_η_y is injective, we conclude that ∑_γ:x→ y f(γ)β_γ(s(x))=0. This implies that if we fix y∈^(0),

    fs(y)=∑_x∈Orb(y)∑_γ:x→ yf(γ)β_γ(s(x))=0,

and hence f∈().

The last part follows from Theorem <ref> and Remark  <ref>.




§ APPLICATIONS




 §.§ Primitive ideals
 In this section, we prove that every primitive ideal of  is the annihilator of a single induced representation. Recall that an ideal is (left) primitive if it is the annihilator of a simple (left) module.


Let 𝒢 be an ample groupoid and 𝒪 a 𝒢-sheaf of rings. If I is a primitive ideal of , then there exists x∈^(0) and M a left B_x-module such that I=(_x(M)).



By Theorem <ref> and Remark <ref>, there exists ℳ a 𝒢-sheaf of 𝒪-modules such that I=(Γ_c(𝒢,ℳ))=⋂_y∈^(0)(_y(ℳ_y)) and  is a simple -module. The latter implies that there exists x∈^(0) such that ℳ_x≠ 0 . We take M=ℳ_x with the left B_x-module structure given by (<ref>). Clearly I(_x(M)).



Suppose now that I≠(_x(M))=:J_x. Then there exists s∈ such that J_xs≠ 0. Since  is simple and J_xs is a non-trivial submodule, we have that J_xs=. Consider now m∈ M∖{0} and let t∈ be such that t(x)=m (such t exists by Proposition <ref>). Then there exists f∈ J_x such that t=fs, and therefore,

    m=t(x)=fs(x)=∑_y∈Orb(x)∑_γ:y→ x f(γ)β_γ(s(y)).

On the other hand, since f∈ J_x and for any y∈Orb(x) we have β_η_y^ -1(s(y))∈ℳ _x, using (<ref>) and Lemma <ref> (and recalling that η_x=x), we obtain that

    0=∑_γ:y→ xf(γ)γη_y·β_η_y^ -1(s(y))=∑_γ:y→ x f(γ)β_γ(s(y)).

This way,

    0≠ m=∑_y∈Orb(x) 0=0,

which is a contradiction.




 §.§ Simple modules
 
We now prove that the induction of modules preserves simplicity.


Let 𝒢 be an ample groupoid, 𝒪 a 𝒢-sheaf of rings and x∈^(0). If M is a simple B_x-module, then _x(M) is a simple -module.



For w∈_x(M)∖{0}, we have to prove that · w=_x(M). Due to the decomposition given in (<ref>), we can write w=∑_i=1^k 1_η_y_i⊗ m_i, where y_i∈Orb(x) for all i, y_i≠ y_j if i≠ j, and m_i∈ M∖{0} for all i. Since ^(0) is Hausdorff, we can find U^(0) open set containing only y_1 among the y_i. Using Equation (<ref>), we can check that χ_U· w=1_η_y_1⊗ m_1. Then, we may assume without loss of generality that w=1_η_y⊗ m for some y∈Orb(x) and some m∈ M∖{0}. Also, it is sufficient to prove that given z∈Orb(x) and m'∈ M, there exists f∈ such that f· w=1_η_z⊗ m'.

We now fix z∈Orb(x) and m'∈ M. Since M is simple, there exists a∈ B_x such that a· m=m'. We can write a=∑_i=1^k a_iδ_i, where a_i∈_x and δ_i∈^x_x for all i. For each i=1,…,k, we take U_i a compact-open bisection containing δ_i and s_i∈ such that s_i(x)=a_i, see Proposition <ref>. We also take compact-open bisections U_y and U_z such that η_y∈ U_y and η_z∈ U_z respectively. Define f=∑_i=1^kχ_U_z*s_iχ_U_i*χ_U_y^-1. Arguing similarly to what was done in the proof or Theorem <ref> to prove Equation (<ref>), starting from (<ref>), we see that

    f· 1_η_y⊗ m     =∑_v∈Orb(x)∑_ζ:y→ v1_η_v⊗α_η_v(f(ζ))η_vζη_y · m  
       =1_η_z⊗∑_i=1^k∑_ζ:y→ z 
    ζ=η_zδ_iη_yα_η_z(χ_U_z*s_iχ_U_i*χ_U_y^-1(ζ))η_zζη_y· m  
       =1_η_z⊗∑_i=1^k α_η_z(α_η_z(s_i((δ_i))))δ_i· m 
       =1_η_z⊗∑_i=1^k a_iδ_i· m 
       =1_η_z⊗ a· m
       =1_η_z⊗ m'.




We now show that the module M in Theorem <ref> above can be chosen to be simple under some strong hypotheses on the rings B_x.  Let J(S) denote the Jacobson radical of a ring S. A ring S is called a left max ring if each non-zero left S-module has a maximal (proper) submodule.  For example, any Artinian ring  S is a left max ring.  Indeed, if M≠ 0, then J(S)M≠ M by nilpotency of the Jacobson radical.  But M/J(S)M is then a non-zero S/J(S)-module and every non-zero module over a semisimple ring is a direct sum of simple modules and hence has a simple quotient.  Thus M has a maximal proper submodule.  A result of Hamsher <cit.> says that if S is commutative, then S is a left max ring if and only if J(S) is T-nilpotent  (e.g., if J(S) is nilpotent) and S/J(S) is von Neumann regular ring. We now prove an analog of the Effros-Hahn conjecture for groupoid algebras with coefficients in a sheaf if all the isotropy skew group rings are left max rings.


Let 𝒢 an ample groupoid and 𝒪 a 𝒢-sheaf of rings, with the isotropy skew group rings B_x being left max rings for all x∈𝒢^(0).  Then the primitive ideals of  are exactly the ideals of the form (_x(M)), where M is a simple B_x-module.


By Theorem <ref> it suffices to show that any primitive ideal I is of the form (_x(M)) where M is a simple B_x-module.
By Theorem <ref> we may assume that our simple module with annihilator I is of the form Γ_c(,ℳ) for some -sheaf ℳ of 𝒪-modules. Let x∈𝒢^0 with ℳ_x≠ 0.  We already know from the proof of Theorem <ref> that I = (_x(ℳ_x)).  Let N be a maximal B_x-submodule of ℳ_x (which exists by assumption on B_x) and let J=(_x(ℳ_x/N)).  Since ℳ_x/N is simple, it suffices to show that J=I.  Clearly, I⊆ J (by Lemma <ref>) since (ℳ_u)⊆(ℳ_u/N).  So it suffices to show that J annihilates Γ_c(,ℳ).

Suppose that this is not the case.  Then there exists s∈Γ_c(,ℳ) with Js≠ 0.  Since J is an ideal and Γ_c(,ℳ) is simple, we deduce Js=Γ_c(,ℳ).   Let m∈ℳ_x∖ N (using that N is a proper submodule) and let t∈Γ_c(,ℳ) with t(x)=m.  Then t=fs with f∈ J.  Let us compute

    m=t(x)=(fs)(x) = ∑_v∈𝒪_x∑_γ v→ xf(γ)β_γ(s(v)).


Let us fix v∈𝒪_x, fix γ_v x→ v and set γ_x=x.  Then by the assumption f∈ J and Lemma <ref>, we have (since β_γ_v( s(v)∈ℳ_x) that 

    ∑_γ v→ xf(γ)β_γ s(v)=∑_γ v→ x f(γ)β_γγ_v(β_γ_v s(v))∈ N.
  We deduce from (<ref>) that m∈ N, which is a contradiction. It follows that J annihilates Γ_c(,ℳ) and so I=J.


A result of Park shows that if R is a unital ring and G is a group acting on R by automorphisms, then the skew group ring R⋊ G is Artinian if and only if R is Artinian and G is finite, see <cit.>.  This leads to the following corollary to Theorem <ref>.



Let  be an ample groupoid with finite isotropy groups and let 𝒪 be a -sheaf of Artinian rings.     Then the primitive ideals of  are precisely the annihilators of modules induced from simple modules of isotropy skew group rings.


Under these hypotheses each isotropy skew group ring is Artinian and hence left max.  The result follows from Theorem <ref>.




 §.§ von Neumann regularity of 


In this section we prove that  is von Neumann regular (a ring R is von Neumann regular if a∈ aRa for all a∈ R) if and only if  is a sheaf of fields. One interesting consequence of this is that the zero section has closed image, which will be used in results studying the algebraic properties of .

To show that when 𝒪 is a sheaf of fields the zero section is closed we need first a sheaf theoretic result. Recall that R^× denotes the group of units of a ring R.


Let 𝒪 be a -sheaf of rings. Then 𝒪^×=⋃_x∈^(0)𝒪_x^× is open and the inversion map on 𝒪^× is continuous.


Let r∈𝒪_x^× with inverse r'.  Then there are neighborhoods U,U' of x in ^0 and sections s U→𝒪, s' U'→𝒪 with s(x)=r and s'(x)=r'.  Without loss of generality, we may assume that U=U', or else we can replace them by U∩ U'.  Since r'r=rr'=1_x, there is a neighborhood W of x with W⊆ U such that (s'∗ s)|_W=(s∗ s')|_W=χ_W and so r∈ s(W)⊆𝒪^×. As s(W) is open we deduce 𝒪^× is open. Note that on s(W), inversion is given by s'∘ p where p𝒪→^(0) is the projection.  Thus inversion is continuous.



Let 𝒪 be a -sheaf of fields.  Then the image of the zero section is closed.


The complement of the image of the zero section is 𝒪^× and so the result follows from Lemma <ref>.


We also use Lemma <ref> in the characterization of von Neumann's regularity of , as we see below. Recall that a unital ring R is indecomposable if it has no central idempotents except 0 and 1.  For example, fields and integral domains are indecomposable.  The special case of the following result for sheaves of rings on a compact totally disconnected space can be found in <cit.>.



Let 𝒪 be a -sheaf of indecomposable commutative rings.  Then  is von Neumann regular if and only if 𝒪_x is a field for all x∈^(0).


If 𝒪 is a sheaf of fields and s∈, then U=(s) = s^-1(𝒪^×) is compact open (as 𝒪^× is open by Lemma <ref>) and s U→𝒪^×.  By continuity of inversion on 𝒪^×, we deduce that s' U→𝒪^× given by s'(x) =s(x) is continuous and, by construction, it has support U. Thus s'∈.  Trivially, s∗ s'∗ s=s and so  is von Neumann regular. 

Conversely, suppose that  is von Neumann regular.  Let 0_x≠ r∈𝒪_x.  Then there exists s∈ with s(x)=r.  Choose s'∈ with s∗ s'∗ s=s. Note that if r'=s'(x), then rr'r=r and so rr' is a non-zero idempotent. But since 𝒪_x is indecomposable, this implies rr'=1_x.  Thus r∈𝒪_x^×.


Note that the Pierce sheaf associated to a commutative ring A is a sheaf of indecomposable commutative rings over the Pierce spectrum of A and hence the construction used in <cit.> (see Section <ref>) to build a groupoid algebra with coefficients in a sheaf of ring from a skew inverse semigroups ring A⋊ S will produce a sheaf of fields when A is commutative and von Neumann regular, see <cit.>.




 §.§ The centralizer of  in 


Maximal commutativity of  will play a key role in describing primitivity and simplicity of . It is also a much-studied concept in the theory of partial skew rings, see <cit.>. Of course, 𝒪 should be a -sheaf of commutative rings in order for  to be commutative. In this section, we characterize (under suitable hypotheses on 𝒪) the centralizer of  in , denoted by C_(), and give sufficient and necessary conditions for  to be maximal commutative. We also get a Generalized Uniqueness Theorem for , similar to what is found for Steinberg algebras in <cit.>.

We begin with an important observation that is true for any -sheaf of rings (not necessarily commutative).


Let  be an ample groupoid and 𝒪 a -sheaf of rings.   If f∈ centralizes , then (f)⊆Iso().  
In particular, if  is Hausdorff and the zero section has closed image, then f∈Γ_c(Int(Iso()),𝒪).


 If (γ)≠(γ) then, by the Hausdorff property of ^(0), there is a compact open subset U of ^(0) with (γ)∈ U and (γ)∉ U.  Therefore f(γ) = f∗χ_U(γ)=χ_U∗ f(γ) = 0. Thus, f is supported on Iso().  If, in addition,  is Hausdorff, then f is continuous and hence, since the zero section is closed, (f) is open.  Thus f∈Γ_c(Int(Iso()),𝒪).


We immediately obtain the following corollary.


Let  be an effective Hausdorff ample groupoid and let 𝒪 be a sheaf of commutative rings with the zero section closed.  Then  is a maximal commutative subring of .



Taking  as the constant sheaf of Example <ref>, the above corollary recovers <cit.>.




We will impose a few extra conditions on the sheaf 𝒪 in order to obtain a characterization of the centralizer.
The first concept we need to introduce is the following.
 
 Let 𝒪 be a -sheaf of rings. We define 𝒪 = {γ∈Iso()|α_γ(a)=a, ∀ a∈𝒪_(γ)}.  Note that 𝒪 is a subgroupoid of Iso() and Int() is an ample subgroupoid. Therefore, we can consider the ring Γ_c(Int(),), which can also be described as the additive subgroup of  generated by sχ_U, where U is a compact-open bisection contained in Int().
 













































































Let  be an ample groupoid and 𝒪 a -sheaf of commutative rings. Then

    Γ_c(Int(),){f∈|(f)} C_().

If  is Hausdorff and the zero section has closed image, then the first inclusion is an equality. Moreover, if 𝒪 is -sheaf of integral domains, then the second inclusion is an equality.



The first inclusion is immediate. For the second inclusion, suppose first that f∈ is such that (f). Then, for g∈, we have that

    f*g(γ)=
    f(γ)α_γ(g((γ))),    if γ∈(f) 
    
    0,    else,

and

    g*f=
    g((γ))α_(γ)(f(γ)),    if γ∈(f) 
    
    0,    else.

For γ∈(f), we have that

    f(γ)α_γ(g((γ)))     =f(γ)g((γ)) 
       =g((γ))f(γ)
       =g((γ))α_(γ)(f(γ)),

and hence g*f=f*g. Since g was arbitrary, f∈ C_().

When  is Hausdorff, an element f∈ is continuous. If, moreover the zero section has closed image, then (f) is an open subset of , from which we get the inclusion {f∈|(f)}Γ_c(Int(),).

Now, suppose that 𝒪 is -sheaf of integral domains. For an element f∈ C_(), we claim that f is supported on 𝒪. We already know that f is supported on Iso() by Proposition <ref>. 
Suppose that γ∈Iso() and f(γ)≠ 0.  Put x=(γ)=(γ).  For a∈𝒪_x, let s be a section in  with s(x)=a (such a section exists by Proposition <ref>).  Then,

    f(γ)α_γ(a) = f∗ s(γ) = s∗ f(γ) = af(γ).
  Since 𝒪_x is an integral domain and f(γ)≠ 0, we deduce that α_γ(a)=a.





We now characterize maximal commutativity of , without any assumption of Hausdorffness in the groupoid.



Let  be an ample groupoid and 𝒪 a -sheaf of commutative rings. If  is a maximal commutative subring of , then Int()=^(0). If moreover 𝒪 is a -sheaf of integral domains and Γ_c(Int(),)={f∈|(f)}, then the converse is also true.



Suppose first that  is maximal commutative and let U⊆𝒪 be compact open.  We must show that U⊆^(0).  
We have that χ_U centralizes  by Lemma <ref> and hence, by maximality, χ_U∈.  Therefore,  U⊆^(0) and  Int()=^(0) as required. To get the equality Γ_c(Int(),)={f∈|(f)}, we observe that  Γ_c(Int(),){f∈|(f)} and apply Lemma <ref>.

The converse follows immediately from the hypothesis and Lemma <ref>.





 Recall that when 𝒪 is the constant sheaf Δ(R) of Example <ref>, Γ_c(𝒢,Δ(R)) is the usual groupoid algebra of <cit.> and Γ_c(𝒢^(0),Δ(R)) is the so called diagonal sub-algebra. In this case, we have that Δ(R) = Iso(𝒢), and so the condition Int(Δ(R))=^(0) says that 𝒢 is effective. We conclude that Γ_c(𝒢^(0),Δ(R)) is maximal commutative in the usual groupoid algebra Γ_c(𝒢,Δ(R)) if, and only if, 𝒢 is effective and Γ_c(𝒢^(0),Δ(R)) =  {f∈Γ_c(𝒢,Δ(R))|(f)Iso(𝒢) }.


 In the case of Hausdorff groupoids, and sheaves such that the zero section is closed, the criteria given in Proposition <ref> for maximal commutativity can be simplified, as we show below. 
 


Let  be a Hausdorff ample groupoid and 𝒪 a -sheaf of integral domains such that the zero section is closed (eg.  is a -sheaf of fields).  Then,  is a maximal commutative subring of  if and only if Int(𝒪) = ^(0).  In particular, if  is effective, then  is a maximal commutative subring.



When  is Hausdorff, an element f∈ is continuous. If, moreover the zero section has closed image, then (f) is an open subset of , from which we get the inclusion {f∈|(f)}Γ_c(Int(),). By Lemma <ref>, we then get that  is maximal commutative if and only if =Γ_c(Int(𝒪),𝒪).  But this is trivially equivalent to ^(0)=Int(𝒪).








[Generalized Uniqueness Theorem]
 Let  be an ample groupoid and 𝒪 a -sheaf of commutative rings. A ring homomorphism π:→ A is injective if and only if π|_C_() is injective.



The “only if” part is immediate. For the “if” part, we notice that since  is a skew inverse semigroup ring by the results of Section <ref>, we can use the second part of the proof of <cit.> to show that if I is a non-zero ideal of , then I∩ C_()≠{0} from which the result follows.


In the Hausdorff case, we can use Lemma <ref> and Corollary <ref> to obtain the following.


Let  be a Hausdorff ample groupoid and 𝒪 a -sheaf of integral domains such that the zero section is closed (eg.  is a -sheaf of fields). A ring homomorphism π:→ A is injective if and only if π|_Γ_c(Int(),) is injective. If in addition Int()=^(0) (eg.  is effective), π is injective if and only if π|_ is injective.






 §.§ Primitivity of 


In this section, we give some necessary conditions and some sufficient conditions for  to be left primitive. Under certain hypotheses on the sheaf, the condition will be both necessary and sufficient. Recall that a ring is said to be left primitive if it has a faithful simple left module.  The following theorem generalizes a result of <cit.> for Steinberg algebras.


Let  be an ample groupoid and 𝒪 a 𝒢-sheaf of rings.  



    
  * If  is left primitive, then there is an orbit Orb(x) such that, for f∈, f|_ (Orb(x))=0 implies f=0.
    
  * Suppose that there is an orbit Orb(x) with B_x left primitive and such that f|_ (Orb(x))=0 implies f=0 for f∈. Then  is left primitive.



Let M be a faithful simple left -module. 
There exists a -sheaf of -modules ℳ such that M= by the Disintegration Theorem. By Theorem <ref>, there exists x∈^(0) such that {0}=(M)=(_x(ℳ_x)). We claim that f|_ (Orb(x))=0 implies f=0 for f∈.  Indeed, f annihilates _x since if (γ)=x and a∈𝒪_(γ), then f· a1_γ = ∑_(β)=(γ) f(β)α_β(a)1_βγ=0 as (γ)∈Orb(x).  Thus f annihilates M. Since {0}=(M)=(_x(ℳ_x)), we conclude that f=0. 

For (2), let M be a faithful simple left B_x-module and consider N=_x(M) the induced module. That N is a simple left -module follows from Theorem <ref>.

We now prove that N is faithful. For that, we fix f∈∖{0}. By assumption, there exists γ∈(f) such that (γ)∈Orb(x). We choose elements η_y for y∈Orb(x) as per (<ref>). Consider the following element of B_x:

    a=∑_ζ:(γ)→(γ)α_η_(γ)(f(ζ))η_(γ)ζη_(γ),

and notice that a≠ 0, since the coefficient of η_(γ)γη_(γ) is α_η_(γ)(f(γ)), which is non-zero. Since M is faithful, there exists m∈ M such that a· m≠ 0. By (<ref>), the component of f· 1_η_(γ)⊗ m corresponding to the coordinate given by 1_η_(γ) is

    ∑_ζ:(γ)→(γ)1_η_(γ)⊗α_η_(γ)(f(ζ))η_(γ)ζη_(γ)· m = 1_η_(γ)⊗ a· m≠ 0.

Hence f· 1_η_(γ)⊗ m≠ 0, which proves the faithfulness of N.


The first condition in Theorem <ref> implies that if  is left primitive, then  has a dense orbit.


Let 𝒢 be an ample groupoid and 𝒪 a 𝒢-sheaf of rings. If  is left primitive, then there exists x∈^(0) such that Orb(x)=^(0).


Let Orb(x) be as in Theorem <ref>(1). If ∅≠ U is a compact open subset of ^(0), then χ_U≠ 0 and so there exists y∈Orb(x) with χ_U(y)≠ 0, that is, y∈ U.  We conclude that Orb(x) is dense.



Notice that, in the case of the constant sheaf (as in Example <ref>) the above result generalizes <cit.> to allow for usual groupoid algebras over possibly non-commutative rings (as opposed to fields).




Let 𝒢 be a Hausorff ample groupoid and 𝒪 a 𝒢-sheaf of rings such that the zero section has closed image. If there exists x∈^(0) such that Orb(x)=^(0) and B_x is left primitive, then  is left primitive.



If 0≠ f∈ then, since f is continuous and the image of the zero section is closed, f is open, whence ( f) is open.  By density of Orb(x), we conclude there exists γ∈(Orb(x))∩(f).  The result now follows from Theorem <ref>(2).



Next, we give necessary and sufficient conditions for  to be left primitive in the case that 𝒪 is a sheaf of fields and  is a maximal commutative subring.



Suppose that 𝒪 is a -sheaf of fields and  is a maximal commutative subring.  Then  is left primitive if and only if ^(0) has a dense orbit.


Corollary <ref> shows that the condition is necessary.  For the converse, let x have a dense orbit, B_x be as usual and let M be any simple B_x-module (since B_x is unital, these exist).  We verify that _x(M) is a faithful left -module.   It is simple by Theorem <ref>.  

Let I be the annihilator of _x(M).  Recall from <cit.> that  is a skew inverse semigroup ring ⋊ S for an appropriate inverse semigroup S.  By <cit.>, since  is maximal commutative, if I≠ 0, then there exists 0≠ s∈ I∩.  Since s is continuous and the orbit of x is dense, 0≠ s(y) for some y∈Orb(x).   Let γ x→ y and let U be a compact open bisection containing γ.  Then χ_U∗ s∗χ_U∈ I∩ and χ_U∗ s∗χ_U(x) =  α_γ(s(y))≠ 0.  Thus without loss of generality, we may assume that s(x)≠ 0.  Since s(x) is a unit of B_x, s(x)m≠ 0 for all m∈ M∖{0}.  So if 0≠ m∈ M, then s· x⊗ m = s(x)x⊗ m =xα_x(s(x))⊗ m= xs(x)⊗ m= x⊗ s(x)m≠ 0, contradicting that s∈ I.  Thus I=0.  This completes the proof.



Let  be a Hausdorff ample groupoid and 𝒪 a -sheaf of fields.  Assume that Int(𝒪)=^(0), e.g., if  is effective.  Then,  is left primitive if and only if ^(0) has a dense orbit.  





 §.§ Semiprimitivity

Recall that a ring is semiprimitive if its Jacobson radical is zero.  Equivalently, it is semiprimitive if it has a faithful semisimple module.  There are many open questions about the semiprimitivity of group rings and skew group rings, in particular, it is still unknown if a group algebra over a field of characteristic 0 is necessarily semiprimitive. So we shall endeavor to understand things modulo this situation.  The results of this section generalize the semiprimitivity results for Steinberg algebras from <cit.>.

Recall that a subset D^(0) is invariant if for all γ∈, (γ)∈ D if and only if (γ)∈ D. If X is an invariant subset of ^(0), then |_X will denote the restriction of  to X. 


Let  be an ample groupoid and 𝒪 a 𝒢-sheaf of rings.  
Suppose that there is an invariant subset X with B_x semiprimitive for some x in each orbit of X and such that f|_|_X=0 implies f=0 for f∈. Then  is semiprimitive.


Let T be a set of orbit representatives of X with B_x semiprimitive for all x∈ T. Let M_x be a faithful left semisimple B_x-module and consider N=⊕_x∈ T_x(M_x). Then N is a semisimple left -module by Theorem <ref>.

We now prove that N is faithful. For that, we fix f∈∖{0}. By assumption, there exists γ∈|_X such that f(γ)≠ 0.  Say that (γ)∈Orb(x) with x∈ T. Again, we use the notation of (<ref>). Consider the following element of B_x:

    a=∑_ζ:(γ)→(γ)α_η_(γ)(f(ζ))η_(γ)ζη_(γ),

and notice that a≠ 0, since the coefficient of η_(γ)γη_(γ) is α_η_(γ)(f(γ)), which is non-zero. Since M_x is faithful, there exists m∈ M_x such that a· m≠ 0. By (<ref>), the component of f· 1_η_(γ)⊗ m corresponding to the coordinate given by 1_η_(γ) is

    ∑_ζ:(γ)→(γ)1_η_(γ)⊗α_η_(γ)(f(ζ))η_(γ)ζη_(γ)· m = 1_η_(γ)⊗ a· m≠ 0.

Hence f· 1_η_(γ)⊗ m≠ 0, which proves the faithfulness of N.

 

Let 𝒢 be a Hausorff ample groupoid and 𝒪 a 𝒢-sheaf of rings such that the zero section has closed image (e.g., a sheaf of fields). If there exists a dense invariant subset X⊆^(0) with  B_x semiprimitive for some x in each orbit, then  is semiprimitive.


If 0≠ f∈ then, since f is continuous and the image of the zero section is closed, f is open, whence ( f) is open.  By density of X, we conclude there exists γ∈|_X∩(f).  The result now follows from Theorem <ref>.



Suppose that 𝒪 is a -sheaf of fields and  is a maximal commutative subring.  Then  is semiprimitive.


For x∈^(0), let B_x be the skew group ring, as usual, and let M_x be any simple B_x-module (since B_x is unital, these exist).  We verify that M=⊕_x∈^(0)_X(M_x) is a faithful semisimple left -module.   It is semisimple by Theorem <ref>.  

Let I be the annihilator of M.  Recall from <cit.> that  is a skew inverse semigroup ring ⋊ S for a suitably chosen inverse semigroup S.  By <cit.>, since  is maximal commutative, if I≠ 0, then there exists 0≠ s∈ I∩.  Suppose that s(x)≠ 0 with x∈^(0).  Since s(x) is a unit of B_x, s(x)m≠ 0 for all m∈ M_x∖{0}.  So if 0≠ m∈ M_x, then s· x⊗ m = s(x)x⊗ m =xα_x(s(x))⊗ m= xs(x)⊗ m= x⊗ s(x)m≠ 0, contradicting that s∈ I.  Thus I=0.  This completes the proof.



Let  be a Hausdorff ample groupoid and 𝒪 a -sheaf of fields.  Assume that Int(𝒪)=^(0), e.g., if  is effective.  Then  is semiprimitive.  






 §.§ Simplicity


Under the hypothesis that we have a sheaf of fields, we give necessary and sufficient conditions for the ring  to be simple. 



 We say that  is minimal if the only open invariant subsets of ^(0) are {0} and ^(0). For the action of ^a on , given in Section <ref>, we say that an ideal I of  is ^a-invariant if for all s∈^a, α_s(D_s^*∩ I) I. Finally,  is ^a-simple if the only ^a-invariant ideals of  are {0} and . 

We aim to show that minimality is equivalent to ^a-simplicity for a large class of 𝒢-sheaves of commutative rings. To see that some hypothesis on 𝒪 is needed, let  be any ample groupoid and R a commutative ring that is not a field.  Let 𝒪 be the constant sheaf associated with R.  Let I be a nonzero proper ideal in R.  Then the set of functions in   that take values in I is a nonzero, proper ^a-invariant ideal that contains no characteristic function χ_V with V≠∅. So minimality does not imply ^a-simplicity in this case.

In what follows, for  an ample groupoid, 𝒪 a -sheaf of rings and U^(0) open, we define the ideal I_U={f∈| f U}.


Let  be an ample groupoid and 𝒪 a -sheaf of rings.

    
  *  If U^(0) is open, then I_U is generated by χ_V with V U compact-open.
    
  *  If I is an ideal of  and U=⋃_χ_V∈ IV, then I_U is generated by χ_V such that χ_V∈ I. In particular, I_U I.



(<ref>) Let f∈ I_U. Since (f) is compact and ^(0) has a basis of compact opens, we can find a compact open set V with (f)⊆ V⊆ U (choose a compact open neighborhood of each x∈(f) contained in U and take the union of a finite subcover).  Then f=fχ_V and so I_U is indeed generated by the χ_V with V⊆ U compact open.

(<ref>) Notice that if V,W are such that χ_V,χ_W∈ I, then χ_V∪ W, χ_V∖ W and χ_V∩ W are all in I. Also, if χ_V∈ I and W⊆ V is compact open, then χ_W = χ_Wχ_V∈ I.  It now follows that if W⊆ U is compact-open, then W⊆ W_1∪⋯∪ W_n with W_i such χ_W_i∈ I for all i=1,…,n, so that χ_W∈ I. It follows that I_U⊆ I by (<ref>).




Let  be an ample groupoid and 𝒪 a -sheaf of indecomposable rings. Then there is a one-to-one correspondence between open subsets of ^(0) and ideals of  generated by idempotents, where if U^(0) is open, we define the ideal I_U={f∈| f U}, and if I is an ideal generated by idempotents, we define U_I=⋃_χ_V∈ I V.  In particular, if 𝒪 is a -sheaf of fields, then there is a bijection between open subsets of ^(0) and ideals of .


If U^(0) is open, then, by Lemma <ref>(<ref>), I_U is generated by the χ_V with V⊆ U compact open. Thus I_U is generated by idempotents and clearly U=⋃_χ_V∈ I_U V since ^(0) has a basis of compact opens.  

Conversely, suppose that I is an ideal generated by idempotents. Since 𝒪 is a sheaf of indecomposable rings, the idempotents of  are the characteristic functions χ_V with V⊆^(0) compact open. It now follows that I⊆ I_U where U=⋃_χ_V∈ I V. By Lemma <ref>(<ref>), we have the reverse inclusion I_U I.

The final statement follows because if 𝒪 is a -sheaf of fields, then  is von Neumann regular by Proposition <ref> and hence every ideal is generated by idempotents (since if aba=a, then ba and a generate the same ideal and ba is idempotent). 
























Let  be an ample groupoid and 𝒪 a -sheaf of rings such that each nonzero ideal I of  contains a characteristic function χ_V with ∅≠ V⊆^(0) compact open.   Then the groupoid  is minimal if and only if  is ^a-simple.


First, suppose that  is minimal.
Let I be a nonzero ^a-invariant ideal and let U=⋃_χ_V∈ IV.  Then by Lemma <ref>(<ref>), we have that I_U is generated by those χ_V in I and hence I_U⊆ I.  Also, by hypothesis, I_U≠ 0 and so U≠∅.  We claim that U is ^a-invariant.  Indeed, if x∈ U and γ x→ y, then by definition there is χ_V∈ I with x∈ V.  Choose W∈^a with γ∈ W.  Then α_W(χ_V)=χ_WVW^-1=χ_Wχ_Vχ_W^-1 is in I and y∈ WVW^-1. Thus y∈ U and so U is invariant.  We conclude that U=^(0) and so =I_U⊆ I, as required.

Conversely, suppose that  is ^a-minimal.  Let U be a nonempty open invariant subset of ^(0).  Then we claim I_U is ^a-invariant.  It suffices to show that if W is a compact open bisection and V⊆ U is compact open, then α_W(χ_V)∈ I_U, as the χ_V with V⊆ U generate I_U by Lemma <ref>(<ref>).  But, as before, α_W(χ_V)=χ_WVW^-1 and WVW^-1⊆ U by invariance.  Thus α_W(χ_V)∈ I_U.  We conclude that I_U=, as I_U≠ 0, and so U=^(0).  



We now specialize the above results. Recall that a commutative ring R with identity is local if the noninvertible elements of R form an ideal 𝔪, which is necessarily the unique maximal ideal of R.  Every field is local and every local ring is indecomposable. 

A sheaf of fields is a sheaf of local rings.  Another example is the following: if X is a locally compact, totally disconnected space and 𝒪 is the sheaf of germs of continuous complex (or real) valued functions on X, then 𝒪 is a sheaf of local rings.  The maximal ideal 𝔪_x of 𝒪_x consists of those germs of functions that vanish at x.  The global sections correspond exactly to continuous functions f on X; the corresponding section sends x to the germ of f at x.  Thus the section corresponding to f vanishes at x∈ X if and only if f(x)=0 and so only the zero section is noninvertible at every point of X.  This motivates the following extension of <cit.>


Let  be an ample groupoid and 𝒪  a -sheaf of rings such  that every nonzero section s∈ is invertible at some x∈^(0).   Then, the groupoid  is minimal if and only if  is ^a-simple. In particular, the statement of the corollary is true if 𝒪 is a sheaf of fields.



Let I be a nonzero ideal of  and 0≠ s∈ I.  Then by hypothesis, there is x∈^(0) such that s(x)∈𝒪_x^× and hence, since 𝒪^× is open by Lemma <ref>, we conclude there is a compact open neighborhood W of x such that s(W)⊆𝒪^×.  Since inversion is continuous on 𝒪^× by Lemma <ref>, there is a section t defined on W with t(y)=s(y)^-1 for all y∈ W.  Then t∈, χ_W=st∈ I and W≠∅.  The result now follows from Proposition <ref>.

The final statement follows because if 𝒪 is a sheaf of fields, then s(x)≠ 0 implies s(x) is invertible.




We remark that if 𝒪 is a -sheaf of commutative local rings, then the hypotheses of Proposition <ref> are equivalent to the hypotheses of Corollary <ref>.  Indeed, the proof of Corollary <ref> provides one implication.  On the other hand, for a -sheaf of local rings, the sections s∈ that are nowhere invertible form an ideal that contains no nonzero characteristic function.  Thus if the hypotheses of  Proposition <ref> apply, then there are no nonzero nowhere invertible sections.









Let  be an ample groupoid and 𝒪 a -sheaf of fields. The following are equivalent:

    
  *   is simple;
    
  *   is ^a-simple and a maximal commutative subring of ;
    
  *   is minimal, Int(𝒪)=^(0) and Γ_c(Int(),)={f∈|(f)}.

If moreover,  is Hausdorff, the above items are also equivalent to:
  
    
  *   is minimal and Int(𝒪)=^(0).




The equivalence between (<ref>) and (<ref>) is given by <cit.>. The equivalence between (<ref>) and (<ref>) follows from Propositions <ref> and <ref>. In the Hausdorff case, the equivalence between (<ref>) and (<ref>) is due to Corollary <ref> and Proposition <ref>.























§ APPLICATIONS TO TOPOLOGICAL DYNAMICS


 Motivated by our results on primitivity and semiprimitivity, described in Corollary <ref> and Theorem <ref>, in this section, we characterize the existence of a dense orbit in an action of an inverse semigroup on a topological space in terms of primitivity of the associated algebra. We will also show that the algebras associated with effective actions are always semiprimitive. 








An important class of actions is formed by topologically free[The terminology “effective” is used in <cit.> but the more usual meaning of effective action is faithful and, indeed, the origin of the term “effective groupoid” is that  is effective if and only if the action of the inverse semigroup of open bisections on ^(0) is faithful.]  actions <cit.>. We recall the definition, in the context of inverse semigroup actions, below. 

[<cit.>]
Let θ=({X_s}_s∈ S,{θ_s}_s∈ S) be an action of an inverse semigroup S on a locally compact Hausdorff space X by partial homeomorphisms. We say that θ is 
topologically free if, and only if, the interior of the set { x ∈ X_s^*|θ_s(x)=x } is equal to 
 
 { x ∈X_s^*  | there is   e ∈E(S)  such that  e≤s  and  x ∈X_e },

for all s ∈ S.


We will see below that the groupoid of germs associated with an inverse semigroup action is the key object connecting properties of the action with properties of the associated algebras. We refer the reader to the second paragraph below Theorem <ref>, or to <cit.>, for the definition of the groupoid of germs. 

The following is proved in <cit.>, <cit.>  and <cit.>.

 Let θ be an action of the inverse semigroup S on the topological space X. Then, θ is topologically free if, and only if, the corresponding groupoid of germs S⋉ X is effective.



 Given an action θ of the inverse semigroup S on the topological space X, the groupoid of germs S⋉ X has a dense orbit if and only if θ has a dense orbit.


 This is immediate, since the unit space of S⋉ X is identified with X, and the groupoid orbit of a point x∈ X is equal to the orbit of x under the action of S, as shown in <cit.>.
 




 If X is locally compact, Hausdorff and second countable, having a dense orbit is equivalent to topological transitivity of the action, see for example <cit.> (or for a special case <cit.>).






Next, we recall how to associate an algebra to a topological inverse semigroup action.
Let R be a unital and commutative ring, and θ=({θ_s }_s∈ S, {X_s}_s∈ S) be an action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X (see <cit.> for details on actions of inverse semigroups). There is a corresponding dual action (see <cit.>) α=({α_s}_s∈ S, {D_s}_s∈ S) of the semigroup S on the R-algebra  of all locally constant, compactly supported, R-valued functions on X where, for each s ∈ S, α_s is the isomorphism from D_s^*= ℒ_c(X_s^*) 
onto D_s= ℒ_c(X_s) given by 

	α_s(f)(x)= {[ f ∘θ_s^*(x)                  x ∈X_s;           0                  x ∉X_s ]..






By <cit.>, the skew inverse semigroup ring, ℒ_c(X)⋊ S, associated to the action α above, is isomorphic to the usual groupoid algebra, denoted by R (S⋉ X), over the groupoid of germs S⋉ X. Recall from Example <ref> that R (S⋉ X) is isomorphic to  Γ_c(S⋉ X,Δ(R)), where Δ(R) is the constant sheaf over S⋉ X. 
More generally, we can consider a S⋉ X-sheaf of rings , which includes the example above as well as the S⋉A-sheaf _A coming from a spectral action of an inverse semigroup S on a unital ring A as in Theorem <ref>. 

We will use the results developed in Section <ref> to relate algebraic properties of the ring Γ_c(S⋉ X,) and topological properties of the action of S on X. 
When the groupoid of germs S⋉ X is not Hausdorff, our results depend on the maximal commutativity of Γ_c((S⋉ X)^(0),𝒪). As we showed in Corollary <ref>, when S⋉ X is Hausdorff this condition is simplified and, in particular, if θ is topologically free then Γ_c((S⋉ X)^(0),𝒪) is maximal commutative. For sufficient conditions for the groupoid of germs to be Hausdorff, see  <cit.> and <cit.> for example.










Let θ=({θ_s }_s∈ S, {X_s}_s∈ S) be a action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X, and let  be a S⋉ X-sheaf of rings. If Γ_c(S⋉ X,) is left primitive then θ has a dense orbit.


By Corollary <ref> the groupoid of germs S⋉ X has a dense orbit. The result now follows from Proposition <ref>.


In the case of a sheaf of fields and Γ_c((S⋉ X)^(0),𝒪) maximal commutative we have a converse of the above result.

 Let θ=({θ_s }_s∈ S, {X_s}_s∈ S) be an action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X, and let  be a S⋉ X-sheaf of fields. Suppose that Γ_c((S⋉ X)^(0),𝒪) is maximal commutative.  Then θ has a dense orbit if, and only if, Γ_c(S⋉ X,) is left primitive.




This follows from the Theorem <ref> and Proposition <ref>.




If S⋉ X is Hausdorff and the action is topologically free, then we do not need the assumption that Γ_c((S⋉ X)^(0),𝒪) is maximal commutative, as we see below.


 Let θ=({θ_s }_s∈ S, {X_s}_s∈ S) be a topologically free action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X, and let  be a S⋉ X-sheaf of fields. If S⋉ X is Hausdorff, then θ has a dense orbit if, and only if, Γ_c(S⋉ X,) is left primitive.




This follows from Corollary <ref> and Proposition <ref>.






 If R is a field then, taking = Δ(R) in the corollary above, we obtain that an action θ on X, whose groupoid of germs is Hausdorff, has a dense orbit if, and only if, ℒ_c(X)⋊ S is left primitive. 



Since Γ_c(S⋉ X,Δ(R)) coincides with the groupoid algebra of <cit.>, the above corollary can also be obtained using Theorem 4.10 in <cit.>. 



Next, we observe that for a sheaf of fields and Γ_c((S⋉ X)^(0),𝒪) maximal commutative, the algebra Γ_c(S⋉ X,) is always semiprimitive.







Let θ=({θ_s }_s∈ S, {X_s}_s∈ S) be an action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X, and let  be a S⋉ X-sheaf of fields. If Γ_c((S⋉ X)^(0),𝒪) is maximal commutative, then Γ_c(S⋉ X,) is semiprimitive.


This follows Theorem <ref>.


As before, if S⋉ X is Hausdorff and the action is topologically free, then we do not need the assumption that Γ_c((S⋉ X)^(0),𝒪) is maximal commutative.


Let θ=({θ_s }_s∈ S, {X_s}_s∈ S) be a topologically free action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X, and let  be a S⋉ X-sheaf of fields. If S⋉ X is Hausdorff, Then Γ_c(S⋉ X,) is semiprimitive.


This follows directly from Proposition <ref> and Corollary <ref>.




Finally, we show that Theorem <ref> can be used to describe the minimality of a topologically free topological action in terms of the simplicity of the associated algebras. In particular, the theorem below should be compared with <cit.>.


 Let θ=({θ_s }_s∈ S, {X_s}_s∈ S) be a topologically free action of an inverse semigroup S on a locally compact, Hausdorff, zero-dimensional space X, and let  be a S⋉ X-sheaf of fields. If S⋉ X is Hausdorff, then θ is minimal if, and only if, Γ_c(S⋉ X,) is simple.


By Proposition <ref>, the groupoid of germs S⋉ X is effective. Hence Int(𝒪)=(S⋉ X)^(0) and the result follows from Theorem <ref>.





§ COMPLEX GROUPOID ALGEBRAS

Let  be an ample groupoid. We let C_c() be the usual ring used to build C^*(), see <cit.>. Since  is ample, C_c() is the span of functions f:→ for which there exists a compact-open bisection U such that (f) U and the restriction f|_U is continuous, where  has the usual topology. Our goal is to prove that C_c(), with the convolution product, is a groupoid ring with coefficients in a -sheaf. In the Hausdorff case, we explicitly build a spectral action (recall the definition of a spectral action from the paragraph above Section <ref>) and show that C_c() is isomorphic to the skew inverse semigroup ring of this action, so that we can apply Theorem <ref>. In the general setting, we build a -sheaf of rings. 




 §.§ The Hausdorff case


Let A:=C_c(^(0)) be the ring of complex valued continuous functions with compact support 
and 𝒢^a be the inverse semigroup of compact-open bisections of  (see Section <ref>). For each U∈𝒢^a, let D_U={f∈ C_c(^(0)):(f)(U)}. Define a spectral action of 𝒢^a on A by α_U:D_U^-1→ D_U, where α_U(f)((γ))=f((γ)) and α_U(f) vanishes outside (U). We then have the following.


With the above conditions, suppose that  is Hausdorff. Then C_c() is isomorphic as a ring to A⋊𝒢^a.



We first build a covariant system for (𝒢^a,A,α). The map θ:A→ C_c() is the usual inclusion of étale groupoid algebras. The map φ:𝒢^a→ C_c() is given by φ(U)=1_U. A straightforward computation using the convolution product shows that (C_c(),θ,φ) is a covariant system for (𝒢^a,A,α). By <cit.>, there is a ring homomorphism π:A⋊𝒢^a→ C_c() such that π(aδ_U)=θ(a)*φ(U) for all U∈𝒢^a and a∈ D_U.

To prove that π is an isomorphism, it suffices to prove that π admits an inverse. Notice that any element f∈ C_c() can be written as

    f=∑_i=1^n f_U_i,

where U_1,…,U_n are pairwise disjoint compact-open bisections such that, for each i=1,…,n, (f_U_i) U_i and the restriction f_U_i to U_i is continuous using that  is Hausdorff. We want to build a map ψ:C_c()→ A⋊𝒢^a. For this, given f∈ C_c() and a decomposition as above, we set

    ψ(f)=∑_i=1^n f_U_i*1_U_iδ_U_i.

We have to prove that ψ is well-defined. The general case follows from the particular case that (f) U, where U is a compact-open bisection with U=⋃_i=1^n U_i. In this case, by the definition of A⋊𝒢^a, we have that

    ∑_i=1^n f_U_i*1_U_iδ_U_i=∑_i=1^n f_U_i*1_Uδ_U=f*1_Uδ_U.

We now prove that ψ=π. First, given f∈ C_c() and a decomposition as above

    π(ψ(f))=∑_i=1^nθ(f_U_i*1_U_i)*φ(δ_U_i)=∑_i=1^n f_U_i*1_U_i*1_U_i=f.

Observing that ψ is a group homomorphism, to show that ψ is a left-inverse for π, it is enough to consider an element aδ_U, where a∈ D_U. In this case,

    ψ(π(aδ_U))=ψ(a*1_U)=a*1_U*1_Uδ_U=aδ_U.

Hence, ψ=π^-1, concluding the proof.









 §.§ The general case


With the previous notation, let us build a -sheaf  as follows. We let A=C_c(^(0)) and =A×^(0)/∼, where the equivalence relation is given by (a,x)∼(a',x') if x=x' and there exists an open neighborhood U of x such that a|_U=a'|_U. The topology on  has as basis the sets

    D(a,U)={[a,x]:x∈ U},

where a∈ A and U is a compact-open set of ^(0). The map p:→^(0) is defined as p([a,x])=x, which is a local homeomorphism by the definition of the topology on . The ring structure on each stalk is the natural one, namely [a,x]+[b,x]:=[a+b,x] and [a,x][b,x]=[ab,x] for a,b∈ A and x∈^(0). Notice that a net {[a_λ,x_λ]}_λ∈Λ converges to [a,x] in  if, and only if, for every open neighborhood U of x, there exists λ_0 such that x_λ∈ U and [a_λ,x_λ]=[a,x_λ] for all λ≥λ_0. With this, it is straightforward to check that conditions (SR1) and (SR2) as in Section <ref> are satisfied. The continuity of the unit section is immediate from the definition of the topology on .

The map α:×_,p→ is given by

    α_γ([a,(γ)])=[a∘∘(|_U),(γ)],

where U is a compact bisection containing γ. Since U is a bisection, the above definition does not depend on a. Also, since the intersection of open bisections is again an open bisection, the definition does not depend on U. To prove that α is continuous, let {(γ_λ,[a_λ,(γ_λ)])}_λ∈Λ be a net converging to (γ,[a,(γ)]) in ×_,p. Fix U a compact-open bisection containing γ, and let V be a compact-open subset of ^(0) containing (γ) and such that V(U). Using that  is continuous, and the description of convergence in  given above, we find λ_0 such that for all λ≥λ_0, we have that γ_λ∈ U, [a_λ,(γ_λ)]=[a,(γ_λ)] and (γ_λ)∈ V. Then, for λ≥λ_0, we have that

    α_γ_λ([a_λ,(γ_λ)])   =α_γ_λ([a,(γ_λ)])
       =[a∘∘(|U)^-1,(γ_λ)]∈ D(a∘∘(|U)^-1,V).

Varying V as above, we obtain a neighborhood basis of [a∘∘(|U)^-1,(γ)], from where we conclude that α is continuous.
Conditions (S1), (S2), and (SR4) of Section <ref> are easy to check. We prove (S3). Let β,γ∈ be such that (β)=(γ) and let [a,(γ)]∈. Also, let U and V be open bisections containing β and γ, respectively. We may assume, without loss of generality, that (U)=(V). Then,

    α_βγ([a,(γ)])     = [a∘∘ (|_UV),(βγ)] 
        = [a∘∘ (|_V)∘∘ (|_U),(β)] 
        = α_β([a∘∘ (|_V),(β)])
        = α_β(α_γ([a,(γ)]).




Let  be an ample groupoid and  the -sheaf constructed above. Then  and C_c() are isomorphic as rings.



First, let f:→ be such that p∘ f= and there exists U compact-open bisection of  such that f|_U is continuous and f|_∖ U=0. Define φ_f:→ by φ_f(γ)=0, if γ∉ U, and φ_f(γ)=a((γ)), if γ∈ U and f(γ)=[a,(γ)]. Since f|_U is continuous, if γ∈ U and f(γ)=[a,(γ)], then there exists an open set V such that γ∈ V U and f(η)=[a,(η)] for all η∈ V. This implies that φ_f|_U is also continuous, and therefore φ_f∈ C_c(). Due to the ring structure on each stalk, we can define a group homomorphism Φ:→ C_c() such that Φ(f)=φ_f, for f a generator of  as above.

On the other hand, given u∈ C_c() such that there exists U compact-open bisection such that u|_U is continuous and (u) U, we can define a_u∈ C_c(^(0)) by a_u(x)=0 if x∉(U) and a_u(x)=u((|_U)^-1(x)) if x∈(U). Then the function ψ_u:→ given by ψ_u(γ)=0 if γ∉ U, and ψ_u(γ)=[a_u,(γ)] is such that ψ_u|_U is continuous and therefore ψ_u∈. Again, we can build a group homomorphism Ψ:C_c()→. Straightforward computations show that Ψ=Φ^-1.

It remains to prove that the above group isomorphism is also a ring isomorphism. We prove that Ψ is multiplicative. Since the maps are already group homomorphisms, due to distributivity, it is enough to consider u,v∈ C_c(G) such that there exist U,V compact-open bisections such that (u) U, (v) V and u|_U and v|_V are continuous. Notice that in this case (u*v) UV, u*v|_UV is continuous and for γ=βρ, where β∈ U and ρ∈ V, we have that (u*v)(γ)=u(β)v(ρ). Keeping the notation of the above paragraphs, we have that

    ψ_u*v(γ)=
    [a_u*v,(β)],    if γ=βρ with β∈ U, γ∈ V
    
    0,    otherwise.

On the other hand,

    (ψ_u *ψ_v)(γ)=
    [a_u,(β)]α_β([a_v,(ρ)]),    if γ=βρ with β∈ U, γ∈ V
    
    0,    otherwise.

Now, if γ=βρ, with β∈ U and ρ∈ V, then

    [a_u,(β)]α_β([a_v,(ρ)])    = [a_u,(β)][a_v∘∘(|_U) ,(β)] 
        = [a_u(a_v∘∘(|_U)),(β)].

Notice that if x∉(UV), then

    a_u*v(x)=0=a_u(x)(a_v∘∘(|_U))(x).

And if x=(γ), where γ=βρ, with β∈ U and ρ∈ V, then

    a_u*v(x)    =(u*v)(γ) 
       =u(β)v(ρ)
       =a_u(x)a_v((ρ))
       =a_u(x)a_v((β))
       =a_u(x)(a_v∘∘(|_U))(x).

It follows that ψ_u*v=ψ_u*ψ_v and hence the multiplicativity of Ψ is proved.





We finish by characterizing simplicity of C_c()



Let  be an ample groupoid and  the -sheaf constructed above. Then, Int(())=^(0) if and only if  is effective.



That  effective implies Int(())=^(0) holds for any -sheaf of rings. For the other implication, let γ∈Int(Iso()) be such that there exists a compact-open bisection U with γ∈ UIso(). Observe that, in this case, ∘(|_U)^-1 is the identity map on (U).
Let a∈ C_c() and ρ∈ U. Then,

    α_ρ([a,(ρ)])=[a∘∘(|_U),(ρ)]=[a,(ρ)],

so that ρ∈(). Hence U() and γ∈Int(())=^(0). It follows that  is effective.




Let  be an ample Hausdorff groupoid. Then, C_c() is simple if and only if  is minimal and effective.



Note that the equivalence of (1) and (2) in Theorem <ref> holds for any -sheaf because it comes from <cit.>. Let  be the sheaf constructed above and consider the isomorphism of Theorem <ref>. Observe that  is a sheaf of commutative rings that satisfies the hypothesis that every non zero section is invertible at some point. By Corollary <ref>, we have that C_c(^(0)) is ^a-simple if and only if  is minimal. If  is effective, then C_c(^(0)) is maximal commutative by Corollary <ref>. On the other hand if C_c(^(0)) is maximal commutative, then  is effective by Proposition <ref> and Lemma <ref>. Hence, we get that C_c() is simple if and only if  is minimal and effective.


[Graph algebras]
Let E be a graph. By means of the graph groupoid _E, it was shown in <cit.> that the Leavitt path algebra L_(E) is dense in the C*-algebra C^*(E). Observe that the algebra C_c(_E) is between L_(E) and C^*(E). Since _E is Hausdorff, using Theorem <ref> and the results of <cit.>, we obtain that the following are equivalent

    
  * L_(E) is simple;
    
  * C^*(E) is simple;
    
  * C_c(_E) is simple.















 §.§ The algebra of continuous functions, with compact support, on the transformation groupoid.


In this subsection, X is always a Hausdorff, locally compact, totally disconnected topological space, and C_c(X) denotes the algebra of all continuous, compactly supported, R-valued functions on X (where R stands for either the real numbers or the complex numbers), with point-wise addition and multiplication. 

It is proved in <cit.> that a partial skew group ring of the form ℒ_C(X) ⋊ G (where ℒ_C(X) stands for the locally constant functions on X) can be seen as the Steinberg algebra associated with the transformation groupoid G ⋉ X. Next, we argue that an analogous result holds when we replace ℒ_C(X) with C_c(X) and the Steinberg algebra with C_c(G ⋉ X). In fact, the outline of the proof is the same as the one of the proof given in <cit.>, and so we will refrain from presenting a whole proof, and instead will only point to the main differences between the two settings. 



For f ∈ C_c(X) we define the support of f by 
    (f)= {x ∈ X  :   f(x)≠ 0}.
 Notice that when dealing with a function f in ℒ_C(X), the set {x ∈ X  :   f(x)≠ 0} is already closed and so it is not necessary to take closure in the definition of support. This is one of the main differences between the C_c(X) and ℒ_C(X) cases. 

Now, let θ=({X_g}_g ∈ G, {θ_g}_g ∈ G) be a partial action of a discrete group G on X, such that X_g is clopen for every g in G. Such action induces an action in the algebra level, as done in <cit.> and <cit.>: For each g in G, consider the ideal D_g:= { f ∈ C_c(X)   :    f   X∖ X_g } in C_c(X), and define α_g: D_g^-1→ D_g by setting α_g(f)=f∘θ_g^-1, for all f ∈ D_g^-1. Then the collection 

    α := ( {D_g}_g ∈ G, {α_g}_g ∈ G )

is an algebraic partial action of G on C_c(X).

Associated with the above partial action we consider the partial skew group ring C_c(X)⋊ G. We also associate to the action θ an étale groupoid, denoted by G ⋉ X, and known as the transformation grupoid (see <cit.>). Let
 
    G ⋉ X:= {(t,x)   :   t ∈ G      x ∈ X_t }.

The inverse of (t, x) ∈ G ⋉ X is 

    (t, x)^-1=(t^-1, θ_t^-1(x)).
 So, the range and source maps r: G ⋉ X →  (G ⋉ X)^(0) and s : G ⋉ X → (G ⋉ X)^(0) are given by r(t,x) =(1,x), and s(t,x)=(1, θ_t^-1(x)) (where 1 denotes the group unit). 

From the above, we have that (s,y), (t,x) ∈ G ⋉ X then (s,y), (t,x) is a composable pair if, and only if, θ_s^-1(y)=x. In this case, we have

    (s,y)(t,x)=(st, y).

Finally, we equip G ⋉ X with the topology inherited from the product topology on G × X. 

We now state the key result in this subsection.



Let θ=({X_g}_g ∈ G, {θ_g}_g ∈ G) be a partial action of a discrete group G over a locally compact, Hausdorff, totally disconnected topological space X, such that each X_g is clopen for all g. Let ({D_g}_g ∈ G, {α_g}_g∈ G) be the corresponding partial action (as defined above) and G ⋉ X be the transformation groupoid associate with θ. Then, C_c(X)⋊ G and C_c(G ⋉ X) are isomorphic as R-algebras. 


The proof is analogous to the proof of <cit.>. One should only take into account that for a general continuous function the set of points where the function does not vanish does not need be closed. 









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