Algebraic actions I. C*-algebras and groupoids

We provide a framework for studying concrete C*-algebras associated with algebraic actions of semigroups: Given such an action, we construct an inverse semigroup, and we introduce conditions for algebraic actions that characterize Hausdorffness, topological freeness, and minimality of the associated tight groupoid. We parameterize all closed invariant subspaces of the unit space of our groupoid, and characterize topological freeness of the associated reduction groupoids. We prove that our groupoids are purely infinite whenever they are minimal, and in the topologically free case, we prove that our concrete C*-algebra is always a (possibly exotic) groupoid C*-algebra in the sense that it sits between the full and essential C*-algebras of our groupoid. As an application, we obtain structural results for C*-algebras associated with, for instance, shifts over semigroups, actions coming from commutative algebra, and non-commutative rings.

1. Introduction 1.1.Context.Algebraic actions of groups form an important class of dynamical systems with deep connections to commutative algebra and operator algebras; they have been studied intensely since their inception by Kitchens and Schmidt in the late 80s (see [49], [66], [31,Chapters 13&14] and the references therein).The theory of their "one-sided" or "irreversible" counterparts, algebraic actions of semigroups, is much less developed.This work is the first in a series of papers in which we systematically study algebraic actions of semigroups from the perspective of operator algebras and groupoids.Here, we consider algebraic actions through the lens of C*-algebra and groupoid theory.Each such algebraic action gives rise to a concrete C*-algebra generated by the Koopman representation for the action together with the left regular representation of the group, and we develop a general framework for studying these C*-algebras using étale groupoids, motivated by natural examples arising from a variety of sources, with the goal of developing a better understanding for their dynamical properties.The observation that algebraic actions of semigroups give rise to new constructions of C*-algebras and groupoids which turn out to be interesting in their own right provides further motivation for our work.
An algebraic action of a semigroup is an action of a semigroup on a group by injective endomorphisms.Until now, the algebraic actions that have been investigated from the point of view of C*-algebra theory fall into two classes.The first class comes from ring theory: The multiplicative monoid of left regular elements of a ring acts by injective endomorphisms on the additive group of the ring, and the associated C*-algebra is the reduced ring C*-algebra of the ring.These were introduced and studied by Cuntz for the ring Z [15], for integral domains with finite quotients by Cuntz and the second-named author in [17], and for general rings by the second-named author in [41].The ring C*-algebras of integral domains with finite quotients that are not fields were proven to be UCT Kirchberg algebras in [17] (following the approach in [15] for the ring Z).For general commutative rings, conditions for pure infiniteness and simplicity were established in [41], but the question of pure infiniteness and simplicity for non-commutative rings was left open.Closely related to these are the C*-algebras associated with actions of congruence monoids on rings of algebraic integers, which have been studied by the authors in the context of boundary quotients of semigroup C*-algebras, and were proven to be UCT The aforementioned actions do not include several fundamental example classes: For instance, actions on solenoids, shifts over semigroups, and algebraic N d -actions on modules over multivariable polynomial rings, which are the one-sided analogues of the actions from Schmidt's classical book [66], are far from the ring-theoretic examples and typically fail to be algebraic dynamical systems in the sense of [7, Definition 2.1].1.2.Motivation.Our motivation is twofold: First, it comes from the observation above that many of the most natural examples of algebraic actions do not fit into any of the existing frameworks, so that structural properties of the associated C*-algebras are not accessible with existing technologies; second, we wanted to find a good construction of an étale groupoid from an algebraic action since the theory of étale groupoids is of interest independent of C*-algebraic considerations.
The groupoid approach to studying a class of C*-algebras is by now a standard one: In order to analyze a class of C*-algebras constructed from some algebraic or combinatorial data, one finds groupoid models for the C*-algebras in question, i.e., one constructs a groupoid from the underlying data and then compares the C*-algebras of the groupoid to the initial C*-algebra of interest.Since there are powerful tools for the structure of groupoid C*-algebras, this often reduces many C*-algebraic questions to problems about groupoids, where ideas from dynamics can be employed.This is the general strategy used here to study the concrete C*-algebras associated with algebraic actions, however we stress that it differs significantly at the technical level from previous works for C*-algebras constructed from graphs, semigroups, or more generally left cancellative small categories.1.3.Novelty.In this paper, we establish a general framework for studying C*-algebras and groupoids associated with general algebraic actions that includes these important classes, where previous approaches are not applicable, as well as the examples from ring theory and algebraic dynamical systems mentioned above.Our idea is to focus on actions of inverse semigroups which naturally arise from algebraic actions of semigroups.On the one hand, the corresponding groupoids give us access to various C*-algebras attached to algebraic actions.At this point, we encounter the interesting phenomenon that essential groupoid C*-algebras or even exotic groupoid C*-algebras naturally enter our discussion.On the other hand, we succeed in establishing structural properties of our groupoids for general classes of algebraic actions, which then have consequences for the C*-algebras we are interested in.In addition, our analysis paves the way for the discovery that the groupoids we associate with algebraic actions are interesting in their own right as they often exhibit surprising rigidity phenomena; this is explained in the subsequent paper [10].Compared to previous approaches, the key novelty of our work is that we do not insist on a strong connection to semigroup C*-algebras because universal C*-algebras arising from the semigroup C*-algebra approach do not provide good models for algebraic actions, as we demonstrate in this paper.
Our approach leads us naturally to a notion of exactness for general algebraic actions (see Definition 4.11), which is a vast generalization of Rohlin's notion of exactness for a single endomorphism (see [65]).This notion is important in our work from the perspective of C*-algebras and groupoids, but it also seems natural from the dynamical point of view.1.4.Overview of results.Let us now explain the main construction of this paper (see § 3) and our results on groupoids.Given any algebraic action σ : S ↷ G (see Definition 2.1), we construct an inverse semigroup as follows: Let I σ be the inverse semigroup of partial bijections of G generated by the endomorphisms implementing the action of S together with the translations for the action of G on itself (see Definition 3.5).The action of S on G generates a distinguished family of subgroups of G, which we call S-constructible subgroups.The collection of non-zero idempotents of I σ is then equal to the collection of cosets for these subgroups.This collection of cosets defines a topology on G, and the resulting completion is a compact, totally disconnected space ∂ E, which carries a natural action of I σ .The groupoid we build from the algebraic action σ is then given by the associated transformation groupoid G σ := I σ ⋉ ∂ E, which coincides with the tight groupoid of I σ in the sense of Exel [21,22].∂ E and its I σ -action have natural interpretations from the perspective of C*-algebras: There is a canonical faithful representation of I σ by partial isometries on ℓ 2 (G), and the C*-algebra generated by this representation is precisely the C*-algebra A σ generated by the Koopman representation of S and the left regular representation of G. Now the Gelfand spectrum of the commutative C*-algebra D σ of A σ generated by the projections coming from idempotents in I σ is canonically homeomorphic to ∂ E. Furthermore, under Gelfand duality, the I σ -action corresponds to the conjugation action by the partial isometries representing I σ on ℓ 2 (G).As a next step, we carefully analyse our groupoids G σ and, among other things, establish the following general structural results (see § 4): • We completely characterize when G σ is Hausdorff, topologically free, or minimal in terms of the algebraic action σ.• We parameterize all closed invariant subspaces of ∂ E = G (0) σ and characterize topological freeness for the corresponding reduction groupoids, again in terms of σ.
• We prove that G σ is purely infinite as soon as it is minimal.The third point can be reformulated by saying that as soon as G σ is minimal, it satisfies comparison in the sense of [33,51], which is an important property in the classification programme for C*-algebras as well as in the study of topological full groups (see Question 4.21 and the discussion surrounding it).We then turn to the question of in what sense G σ is a groupoid model for our concrete C*-algebra A σ .First, we prove that there is always a canonical, surjective *-homomorphism C * (G σ ) → A σ that is an isomorphism at the level of canonical commutative subalgebras.Thus C * (G σ ) can be viewed as a universal model for A σ .At the level of concrete C*-algebras, it turns out that, interestingly, the essential C*-algebra C * ess (G σ ) of G σ often provides the best approximation for A σ .This contrasts with the case of C*-algebras associated to left cancellative small categories where the concrete C*-algebra is typically a quotient of the reduced C*-algebra of the groupoid model, see [46].In § 5, we use recent results by Christensen and Neshveyev on induced representations for groupoid C*-algebras [12] to prove that under mild assumptions, the concrete C*-algebra A σ surjects onto the essential C*-algebra of G σ , and injectivity of this map is characterized by amenability of the homomorphic group image of the inverse semigroup I e generated by the endomorphisms coming from S. In the case where G σ is Hausdorff, reduced and essential groupoid C*-algebras coincide, so that we obtain criteria when A σ can be identified with C * r (G σ ).The essential C*-algebra of an étale groupoid, as defined in [25,37], is an innovation at the heart of recent advances in the study of C*-algebras of non-Hausdorff groupoids (see, e.g., [59,25,37,32,46,12,57]).It captures-at the C*-algebraic level-minimality and pure infinitness of the groupoid even in the non-Hausdorff setting.Our identification of A σ with C * ess (G σ ) allows us to deduce results on structural properties of A σ from our above-mentioned analysis of G σ .For instance, under amenability assumptions, we characterize simplicity and pure infiniteness of A σ in terms of the initial action σ : S ↷ G. Interestingly, we also identify criteria when A σ is an exotic groupoid C*-algebra lying properly between the full and reduced C*-algebras of G σ (see Corollary 5.16).Moreover, we compare C * ess (G σ ) and A σ with C*-algebras from the context of semigroup C*-algebras (see § 6) and show that the latter do not provide good models for A σ in general.
Acknowledgments.C. Bruce would like to thank Bartosz Kosma Kwaśniewski for helpful comments on essential C*-algebras of non-Hausdorff groupoids, and Kevin Aguyar Brix and Gavin Goerke for inspiring discussions on non-Hausdorff groupoids and C*-algebras from actions of inverse semigroups.We also thank Julian Kranz for helpful comments which led to an improvement of Lemma 4.26.

Preliminaries
2.1.Algebraic actions of semigroups.This work is centered around the following class of actions, which are the "one-sided" analogues of the algebraic actions of groups from [66].
Definition 2.1.An algebraic action of a semigroup S is an action of S on a discrete group G by injective group endomorphisms, i.e., a semigroup homomorphism σ : S → End (G) such that σ s is an injective group homomorphism G → G for all s ∈ S.
We shall write σ : S ↷ G to denote an algebraic action.When we want to emphasize the acting semigroup, we call σ : S ↷ G an algebraic S-action.
The case where G is Abelian is the most interesting from the point of view of topological dynamics: Remark 2.2.When G is Abelian, an algebraic S-action on G is the same as having an action of S on the compact Pontryagin dual G of G by surjective group endomorphisms.More precisely, if σ : S ↷ G is an algebraic S-action, then we obtain a right action σ : S ↷ G (i.e., a semigroup antihomomorphism from S to the semigroup of continuous homomorphisms G → G) characterized by ⟨σ s (χ), g⟩ = ⟨χ, σ s (g)⟩ for all s ∈ S, χ ∈ G and g ∈ G, where ⟨•, •⟩ : G × G → T is the duality pairing.Moreover, it is easy to see that, for all s ∈ S, σs is surjective if and only if σ s is injective.
Standing assumptions: Assume that σ : S ↷ G is an algebraic S-action.The identity element of G will be denoted by e.We shall always assume that S is a monoid with identity element denoted by 1 and that G is non-trivial.Note that since σ is injective, we have σ 1 = id.We will also always assume that the action σ : S ↷ G is faithful, i.e., σ is injective.This implies that S is left cancellative.
We say σ : S ↷ G is automorphic if σ s ∈ Aut(G) for all s ∈ S; thus, σ : S ↷ G is non-automorphic if there exists s ∈ S such that σ s G ⪇ G (this implies that S is non-trivial).We will primarily be interested in non-automorphic actions.
Let us now introduce the important concept of globalization.Definition 2.3.We say that the algebraic action σ : S ↷ G has a globalization if S can be embedded into a group S and there is a group G containing G together with an algebraic action σ : S ↷ G (which is necessarily automorphic) such that σs | G = σ s for all s ∈ S.There are many large example classes of algebraic actions that admit a globalization.
Example 2.4.Suppose S is left Ore, i.e., cancellative and right reversible (Ss ∩ St ̸ = ∅ for all s, t ∈ S).Then S acts on the group G = S −1 G := lim − →s {G σs → G} by automorphisms, so we obtain a globalization σ : S ↷ S −1 G, where S = S −1 S is the enveloping group of S.More precisely, consider the inductive system G p := G (for p ∈ S, where p ≤ q if q = rp for some r ∈ S), with connecting map for all p, r ∈ S and g ∈ G. Now define σs : G → G as follows: Given g ∈ G q , find q ′ , s ′ ∈ S with q ′ q = s ′ s.Such elements exist because S is right reversible.Now define σs ([g]) := [σ q ′ (g)], where we view σ q ′ (g) as an element of G s ′ .It is straightforward to check that σs is an automorphism of G which satisfies σs | G = σ s for all s ∈ S, so that σ extends to the desired algebraic action σ : S ↷ G .Note that when G is Abelian, the dual action of σ can be constructed explicitly from X = G and α = σ (the underlying space will be given as a projective limit obtained from (X, α)).
Example 2.5.Suppose G ⊆ Q r is torsion-free and of finite rank r ∈ Z >0 .Then S acts by automorphisms on Q ⊗ Z G, so we obtain a globalization by considering the action of the group generated by S on Q ⊗ Z G.
Example 2.6.Let Σ be a non-trivial group.The full S-shift over Σ is the algebraic S-action This action admits a globalization if and only if S can be embedded into a group (say S ), in which case a globalization is given by the group shift σ : S ↷ S Σ, σg (x) h := x g −1 h .
Remark 2.7.Suppose that S ⊆ S is an embedding of S into a group S .Let σ : S ↷ G be an algebraic action with G Abelian.Consider ZS ⊗ ZS G with the natural S -action and the map G → ZS ⊗ ZS G, g → 1 ⊗ g.This is the universal enveloping action of σ with respect to S ⊆ S , in the sense that any enveloping action of σ with respect to S ⊆ S factors through it.As a consequence, G → ZS ⊗ ZS G is injective if and only if there exists a globalization σ : S ↷ G of σ.

2.2.
Étale groupoids and their C*-algebras.In this and the next subsection, we introduce some notation for groupoids and their C*-algebras that will be used throughout this paper.For C*-algebras of non-Hausdorff groupoids, we refer the reader to [14] or [34].For the Hausdorff case, see, e.g., [63] and [67,Part II].
Let G be a (not necessarily Hausdorff) locally compact étale groupoid such that the unit space G (0) is Hausdorff in the relative topology.We let r and s denote the range and source maps on G.A subset B ⊆ G is said to be a bisection if the restrictions r| Here δ γ is given by δ The groupoids constructed in this paper will not always be Hausdorff, and we shall see that the essential C*-algebra, as defined in [25,37], will provide the best model for our concrete C*-algebras.Essential groupoid crossed products are defined in [37,Definition 7.12] for Fell bundles over G. Specializing to the case of the trivial Fell bundle whose fibres are all equal to C, one arrives at the definition of the essential C*-algebra of G, which is the quotient C * ess (G) := C * r (G)/J sing , where J sing ⊆ C * r (G) is the ideal of so-called singular elements (see [37, § 4]).Several characterizations of J sing are given in [37,Proposition 7.18], but we shall not need them here.Basic properties of C * ess (G) are established (in a more general setting) in [37, § 7].If G is Hausdorff, then C * ess (G) agrees with C * r (G).
2.3.Induced representations.We continue with the setup from the previous subsection.Fix x ∈ G (0) .If π is a representation of the group C*-algebra C * (G x x ) on a Hilbert space H π , then the associated induced representation Ind π of C * (G) can be explicitly defined as follows (cf.[12, § 1]): Let x , and where u g is the canonical unitary in C * (G x x ) corresponding to g.Then, H Ind π is a Hilbert space with the obvious linear structure and inner product and Ind π : We shall make use of the following observation in several places below.Proposition 2.8.There is a unitary W : , so that W V = I, which shows that W is surjective.It is now easy to verify (1).□ Remark 2.9 (cf.[12, § 1]).The induced representation Ind λ G x x coincides with π x • π r .
3. C*-algebras and groupoids associated with algebraic actions , where {δ h : h ∈ G} is the canonical orthonormal basis for ℓ 2 (G).We shall call κ σ the Koopman representation associated with the action σ.
Definition 3.1.Suppose σ : S ↷ G is an algebraic action.We let where λ : G → U(ℓ 2 (G)) is the left regular representation of G.
In the following, we will simply write κ for κ σ if the algebraic action σ is understood.
Remark 3.2.The C*-algebra A σ depends only on the image of σ, so if we are only interested in A σ , then there is no loss in generality in assuming faithfulness of σ : S ↷ G.
Let P := G ⋊ S denote the semi-direct product with respect to σ taken in the category of monoids.We identify G and S via the embeddings g → (g, 1) ∈ P and s → (e, s) ∈ P as submonoids of P .
The following explains our name for κ: Remark 3.4.Suppose G is Abelian.Then in terms of the dual action α := σ : S ↷ X = G, κ and A σ can be understood as follows: If µ denotes normalized Haar measure on X, then α is measurepreserving in the sense that µ(W ) = µ(α −1 s (W )) for all Borel subsets W ⊆ X. Hence we obtain an isometric representation of S on L 2 (X, µ) via L 2 (X, µ) → L 2 (X, µ), f → f • σs (for s ∈ S), which is the analogue of the Koopman representation in the group case.Moreover, C(X) acts on L 2 (X, µ) by multiplication operators.This yields a unitary representation of G on L 2 (X, µ) via the canonical identification C * (G) ∼ = C(X).These two representations of S and G together give rise to a representation of P which is unitarily equivalent to λ ⋊ κ via the canonical unitary L 2 (X, µ) ∼ = ℓ 2 (G).
A priori, we have the following description of A σ : We now set out to construct (a candidate for) a groupoid model for A σ .For this, the language of inverse semigroups is very convenient.

3.2.
The inverse semigroup associated with an algebraic action.For background on inverse semigroups, see [39], and for background on their C*-algebras, see [60] and [21,22].Let I G denote the inverse semigroup of all partial bijections of G.We shall now view σ s as a partial bijection of G, so that σ −1 s makes sense as an element of I G ; namely, σ −1 s is the partial bijection σ s G → G given by σ s (g) → g.For each g ∈ G, let t g ∈ I G denote the bijection G → G given by t g (h) = gh for all h ∈ G. Definition 3.5.We let I σ denote the inverse sub-semigroup of I G generated by the endomorphisms σ s for s ∈ S and the translations t g for g ∈ G. Explicitly, we have In the following, we will simply write I for I σ if the algebraic action σ is understood.
There is a canonical faithful representation by partial isometries Λ : I G → PIsom(ℓ 2 (G)) such that, for ϕ ∈ I G with domain dom(ϕ) and h ∈ G, From now on, we shall use Λ to denote the restriction of the above representation to I. It is easy to see that this restriction extends the isometric representation κ, so that Λ : I → PIsom(A σ ) is a representation of the inverse semigroup I in A σ .Now it follows immediately that A σ = span({Λ ϕ : ϕ ∈ I}), so it is reasonable to expect that the structure of A σ is closely related to properties of the inverse semigroup I.This motivates the analysis of the inverse semigroup I.In particular, we wish to compare the C*-algebras associated with I with the C*-algebra A σ .
In the following, let E σ be the idempotent semilattice of I.
Definition 3.6.We let C σ be the smallest family of subgroups of G such that Here, for a subset C ⊆ G, we put σ −1 s C := {h ∈ G : σ s (h) ∈ C}.We are writing σ −1 s C for the set-theoretic inverse image of C under σ s , rather than the more cumbersome notation σ −1 s (C ∩ σ s G), which would be used when viewing σ s as a partial bijection.
Members of C σ are called S-constructible subgroups.In the following, we will simply write C for C σ and E for E σ if the algebraic action σ is understood.Note that Definition 3.8.Let I e be the inverse sub-semigroup of I generated by {σ s : s ∈ S}, i.e., Note that C is the semilattice of idempotents of I e .
Remark 3.9.The elements in I e are partial group automorphisms, i.e., they are group isomorphisms from their domains onto their ranges.
Proposition 3.10.The family C of S-constructible subgroups satisfies the following properties: Proof.(i) follows because C is the idempotent semilattice of I e , and (ii) follows immediately from the definition of C. Note that a direct proof of (i) can be given following the proof of [42,Lemma 3.3].
Let us now develop a standard form for elements of I.
(i) For ϕ ∈ I, we have ϕ ∈ I e if and only if ϕ(e) = e.(ii) Every ϕ ∈ I × is of the form ϕ = t h φt g −1 for some φ ∈ I e and g, h ∈ G.More precisely, if dom(ϕ) = gC for some g ∈ G and C ∈ C, and if ϕ(g) = h, then φ := t h −1 ϕt g ∈ I e .
(ii): We know by Corollary 3.11 that dom(ϕ) = gC for some □ Corollary 3.13.The inverse semigroup I is 0-E-unitary if and only if I e is E-unitary.
Proof.It is easy to see that I e is E-unitary whenever I is 0-E-unitary.Assume I e is E-unitary, and suppose ϕ ∈ I and kD ∈ E × are such that kD ⊆ dom(ϕ) and ϕ| kD = id kD .Let gC = dom(ϕ).Note that kD ⊆ gC, so that k ∈ gC and D ⊆ C. By Proposition 3.12, we can write ϕ = t h φt g −1 for φ ∈ I e with h = ϕ(g) and dom(φ Since I e is E-unitary, it follows that φ = id C , so that (2) implies g = h.Finally, we see that ϕ = t g id C t g −1 = id gC .□ Remark 3.14.If σ : S ↷ G is non-automorphic, then ∅ ∈ E, and we view ∅ as the distinguished zero element of E (in the sense of [16, Definition 5.5.2]).We shall always regard I e as a semilattice without a distinguished zero element, even though I e may contain the trivial subgroup {e}.

3.3.
The partial algebraic action associated with a globalization.For the basics of partial group actions, see, for instance, [20], [30], or [16, § 5.5].If σ : S ↷ G is a globalization for σ : S ↷ G, then the monoid P = G ⋊ S embeds into the group Γ := G ⋊ S .We get an affine action of Γ on G by (g, s).x := gσ s (x); we shall often identify G and S with their images in Γ. Elements of the group ⟨P ⟩ ⊆ Γ are then of the form s −1 , where s i , t i ∈ S, g i ∈ G, and m ∈ Z >0 .
By restricting to the subgroup G ⊆ G , we obtain a partial affine action Γ ↷ G, where for γ = gs ∈ Γ with g ∈ G and s ∈ S , γ acts as follows: The domain of γ is Remark 3.15.Since the action of g ∈ G is given by t g and the action of s ∈ S is given by σ s , we see that In light of the above remark, it is natural to ask for conditions that will ensure there is a well-defined map I × → Γ given by "ϕ → g ϕ s ϕ ".Suppose σ : S ↷ G has a globalization σ : S ↷ G .Consider the following condition: This condition is a kind of joint faithfulness for the partial action of ⟨S⟩ ⊆ S on G.
Proposition 3.16.Assume σ : S ↷ G has a globalization σ : S ↷ G .Then σ : S ↷ G satisfies (JF) if and only if I is strongly 0-E-unitary.If these equivalent conditions hold, then there exists an idempotent pure partial homomorphism g : I × → Γ such that g(ϕ) = g ϕ s ϕ , where g ϕ s ϕ ∈ Γ is associated with ϕ as in Remark 3.15.
Proof.Assume (JF) is satisfied.Let ϕ ∈ I × .By Proposition 3.12, we can write ϕ = t h φt g −1 for some g, h ∈ G and φ = σ −1 for all x ∈ gC.Taking x = g in (3) gives g ϕ σs ϕ (g) = kσ t (g).Plugging this into (3) gives σs ϕ (c) = σt (c) for all c ∈ C, i.e., σt −1 s ϕ (c) = c for all c ∈ C. Now t −1 s ϕ = e by (JF), i.e., t = s ϕ , and k = g ϕ follows from (3).This shows that g : I × → Γ defined by g(ϕ) = g ϕ s ϕ is well-defined.An argument similar to the one above shows that ϕ ∈ E × if and only if (g ϕ , s ϕ ) = (e, 1), and it is easy to see that g is a partial homomorphism.Hence, g is an idempotent pure partial homomorphism, so that I is strongly 0-E-unitary.Now assume that I is strongly 0-E-unitary, so that there exists an idempotent pure partial homomorphism g : I × → Λ for some discrete group Λ. Observe that the map G ⋊ S → I × , (g, s) → t g σ s is injective, so that we may view G ⋊ S as a submonoid in I × .The restriction of g to the copy of G ⋊ S in I × is injective by [16,Lemma 5.5.7], which gives us an embedding Remark 3.15).Since both sides are non-zero, applying g to ( 4) yields s = s −1 Let us now discuss the special case when S is left Ore and G is Abelian.
Example 3.17.Assume S is left Ore and that G is Abelian.We shall write G additively.
For (i), observe that we have αt = βu, so

3.4.
The unit space of the groupoid model.Let us assume in this subsection that our action is non-automorphic, so that E = {gC : C ∈ C, g ∈ G} ∪ {∅} is the idempotent semilattice of our inverse semigroup I.As before, we write E × := E \ {∅}.Define E as the space of characters of E, i.e., non-zero multiplicative maps E → {0, 1} sending ∅ to 0, equipped with the topology of point-wise convergence.
A basis of open sets is given by where gC ∈ E × and {g i C i } ⊆ E × is a finite subset.Without loss of generality we may assume g i C i ⊆ gC for all i.There is a one-to-one correspondence between characters of E and filters on E, i.e., subsets F ⊆ E with the following properties: ∅ / ∈ F; G ∈ F; if gC ∈ F and hD ∈ E with gC ⊆ hD, then hD ∈ F; and if gC, hD ∈ F, then gC ∩ hD ∈ F. This one-to-one correspondence is implemented by the assignment E ∋ χ → F(χ) := {gC ∈ E: χ(gC) = 1}.Definition 3. 19.Let E max denote the characters χ of E for which F(χ) is maximal with respect to inclusion.
In other words, χ ∈ E belongs to E max if and only if we cannot find χ ′ ∈ E with F(χ) ⊊ F(χ ′ ).Definition 3.20.The boundary of E is given by ∂ E := E max .
Following [21,22], characters of E which belong to ∂ E are called tight, and we also call the corresponding filters tight.We briefly recall several notions from [21,22,24].A cover (resp.outer cover ) of a subset F ⊆ E is a subset c ⊆ F (resp.c ⊆ E) such that for each gC ∈ F × := F ∩ E × , there exists hD ∈ c with hD ∩ gC ̸ = ∅.For ϕ ∈ I, let fix(ϕ) := {g ∈ dom(ϕ) : ϕ(g) = g}, and let where we view gC as an element of I using the identification of E with the idempotent semilattice of I.It is shown in [21,22] that χ ∈ E belongs to ∂ E if and only if for every gC ∈ E × with χ(gC) = 1 and every cover c of gC, there exists hD ∈ c such that χ(hD) = 1.To ease notation, let ∪J ϕ := gC∈J ϕ gC.
In particular, c is a cover of the constructible coset gC if and only if Proof."⇒" follows from the characterization of tight characters in [21,22] mentioned above because {gk i D} is a cover of gC.For "⇐", suppose that {h j D j } is a cover of gC.By [58, (4.4)], we may without loss of generality assume that [C : D j ] < ∞ for all j.Set D := j D j .Then [C : D] < ∞, so that C = i k i D for some k i ∈ G.By assumption, there exists i such that χ(gk i D) = 1.Moreover, as gC = j h j D j by Lemma 3.21, there exists j such that gk i D ⊆ h j D j .Hence χ(h j D j ) = 1, as desired.□ Let us now relate characters and filters on E with those on C. As above, a filter on C is a subset F ⊆ C with the following properties: Proof.(i) is true by construction.For surjectivity of Π in (ii), just observe that, given a filter F on C, χ(F)(C) := 1 if C ∈ F and χ(F)(gC) := 0 for all gC ∈ E with gC / ∈ F defines a character of E which satisfies F(χ(F)) = F.The remaining claims in (ii) are easy to see.
(iii) follows from Lemma 3.22.(iv) is true because any two elements of C always have non-empty intersection, as they all contain e ∈ G.It remains to show (v).Π −1 (F max ) ⊆ E max is clear.Given a non-empty basic open set ∂ E(gC; {h i D i }), there exists k ∈ gC \ i h i D i .Hence kD i ∩ h i D i = ∅ for all i.Moreover, by (ii), there exists χ ∈ Π −1 (F max ) with the property that χ(kC) = 1 for all C ∈ F max .It follows that χ ∈ ∂ E(gC; {h i D i }), as desired.□ Remark 3.25.The compatibility condition in Lemma 3.24 (ii) is equivalent to the condition that (g C ) is an element of the projective limit lim ← −C∈F {G/C}.
The following notation will be convenient.Note that χ k exists by Lemma 3.24 (ii).Indeed, we have In the terminology from [21], G σ is the tight groupoid of I.When I is countable, the C*-algebra C * (G σ ) is universal for tight representations of the inverse semigroup I by [21,Theorem 13.3].
As Λ : I → A σ is a representation, the universal property of C * (I) yields a (surjective) *-homomorphism ρ : Proposition 3.30.Assume I is countable and that σ : S ↷ G is non-automorphic.Then the representation Λ : Proof We for any ϕ ∈ I × with g(ϕ) = γ.We have isomorphisms

Properties of the groupoid model
In this section, we study properties of the groupoid G σ attached to the non-automorphic algebraic action σ : S ↷ G.These groupoid properties then translate into properties of the C*-algebras C * r (G σ ) and C * ess (G σ ).(H) For all ϕ ∈ I, there exist constructible cosets If S is torsion-free, G is Abelian, and the dual action σ is mixing, then fix(σ s ) = {e} for all 4.2.Closed invariant subspaces of the boundary.Recall that we introduced the notion of filters on E in § 3.4, and that there is a one-to-one correspondence between characters of E (i.e., elements of E) and filters given by E ∋ χ → F(χ) := {gC ∈ E: χ(gC) = 1}.This bijection restricts to a one-toone correspondence between tight characters and tight filters.Moreover, we defined the map Π from characters of E to characters of C and set F(χ) := F(Π(χ)) := {C ∈ C: Π(χ)(C) = 1} for every χ ∈ E.
We observed that for all χ ∈ E, χ is tight if and only if F(χ) is finitely hereditary.Moreover, for every filter F on C there exists a uniquely determined character χ(F) ∈ E with F(χ(F)) = F.In particular, for a finitely hereditary filter F on C, χ(F) is a tight character.
In order to describe closed invariant subspaces of ∂ E, we need some terminology.
Definition 4.3.Let F C be the set of finitely hereditary filters on C. We define a partial action of I e on F C as follows: Given F ∈ F C and φ ∈ I e , φ.F is defined if there exists C φ ∈ F such that C φ ⊆ dom(φ).In that case, φ.F is defined as the smallest element of Our main result concerning closed invariant subspaces of ∂ E reads as follows: Theorem 4.4.There is a one-to-one correspondence between closed invariant subspaces of ∂ E and For the proof, we first show that closed, G-invariant subspaces of ∂ E are in one-to-one correspondence with ⊆-closed subsets of Proof.Suppose that χ, ω ∈ ∂ E satisfy Π(ω) = Π(χ).Then we claim that ω ∈ {t k .χ:k ∈ G}.Indeed, take a basic open neighbourhood ∂ E(gC; {h i D i }) of ω.Then ω(gC) = 1 and ω(h i D i ) = 0 for all i.Without loss of generality, we can assume that D i / ∈ F(ω) (otherwise replace C by C ∩ D i ).As Π(ω) = Π(χ), this implies that there exists k ∈ G with χ(kC) = 1 while t h .χ(hi D i ) = 0 for all i and all h ∈ G. Therefore, t gk −1 .χ(gC)= χ(kC) = 1 whereas t gk −1 .χ(hi D i ) = 0 for all i.Hence t gk −1 .χ(gC)∈ ∂ E(gC; {h i D i }), as desired.
"⇐": Without loss of generality, we may assume that F(ω) = F(ω) because Lemma 4.5 allows us to replace ω by χ(F(ω)) if necessary.Take a basic open neighbourhood ∂ E(gC; {h i D i }) of ω, with h i D i ⊆ gC for all i.We may assume g = e.Hence ω(C) = 1 and ω(h i D i ) = 0 for all i.We may also assume that D i / ∈ F(ω) for all i (otherwise replace C by C ∩ D i ).Thus [C : D i ] = ∞ for all i.Since F(ω) ⊆ χ∈X F(χ), we can find χ ∈ X with χ(lC) = 1 for some l ∈ G. Without loss of generality assume that l = e, i.e., χ(C) = 1.For each i with ∅ for all i, and thus t h .χ(hi D i ) = 0 for all i.We conclude that t h .χ∈ ∂ E(C; {h i D i }), as desired.□ Proof.Proposition 4.6 implies that for any closed, G-invariant subspace To see that it is also closed, take ω ∈ F −1 (F).Proposition 4.6 implies that F(ω) ⊆ F∈F F. But since F is ⊆-closed, this implies F(ω) ∈ F and thus ω ∈ F −1 (F), as desired.
To see that these maps are inverse to each other, first note that F(F −1 (F)) = F because F is surjective.Moreover, to show F −1 (F(X)) = X, it suffices to show "⊆" as "⊇" is clear.So take ω ∈ ∂ E with F(ω) ∈ F(X).Then F(ω) = F(χ) for some χ ∈ X.It follows that Π(ω) = Π(χ) and thus ω ∈ {t k .χ:k ∈ G} by Lemma 4.5.This shows "⊆".□ For the proof of Theorem 4.4, it remains to show that the one-to-one correspondence in Corollary 4.7 restricts to a one-to-one correspondence between closed invariant subspaces of ∂ E and I e -invariant, ⊆-closed subsets of F C .In other words, we have to show that the I e -action is preserved.This is a consequence of the following observations: Lemma 4.8.
Proof.(i) and (ii) follow from the observations that φ.χ is defined if and only if there exist g ∈ G and C ∈ C with χ(gC) = 1 and gC ⊆ dom(φ), while the latter is equivalent to g ∈ dom(φ) and C ⊆ dom(φ) because φ ∈ I e , and that φ.F(χ) is defined if and only if there exist C ∈ F(χ) with C ⊆ dom(φ).
For (iii), assume that For "⇐", C ∈ F(χ) implies that there exists g ∈ G such that χ(gC) = 1.Hence there exists □ Remark 4.9.Theorem 4.4 reduces the problem of computing all closed invariant subspaces of ∂ E to the study of certain subsets of F C , and the relevant subsets are singled out by two conditions involving the I e -action and set-theoretical properties, but no topology.The question remains whether it is possible to compute closed invariant subspaces more concretely.In principle, I e -invariant and ⊆-closed subsets F of F C are completely determined by the subset F∈F F of C.However, in general, it seems to be a challenge to characterize which subsets of C arise in this way.
Without loss in generality, we may assume that, for every i, ϕ i ϕ −1 i (C ∩ im(ϕ i )) ∩ D is non-empty, so that there exists a constructible coset . By Proposition 3.12, we can write ϕ i = t h i φ i t g −1 i for some g i , h i ∈ G and φ i ∈ I e such that dom(φ i ) ∈ C and dom(ϕ i ) = g i dom(φ i ).Since k i D i ⊆ dom(ϕ i ), we have k i = g i c i for some c i ∈ dom(φ i ) and D i ⊆ dom(φ i ), so that c i D i ⊆ dom(φ i ).By Remark 3.9, φ i is a homomorphism on dom(φ i ), so that , so by [58, (4.3)], we may assume that φ i (D i ) has finite index in C for all i.Since e ∈ C, there exists i such that e x ∩U ̸ = ∅} has empty interior.In general, G is topologically free whenever it is effective (i.e., Iso(G) • = G (0) , where Iso(G) := x∈G (0) G x x is the isotropy bundle of G).The converse holds if G is Hausdorff.Topological freeness is important because of its relationship to the intersection property, see [37, § 7] and [32, § 7.1].
We now turn to topological freeness for the groupoids I ⋉ X, where X ⊆ ∂ E is a closed, invariant subset.Remark 4.12.The group C∈C C is the biggest subgroup G c of G which is invariant under σ s for all s ∈ S such that σ s | Gc is surjective for all s ∈ S. Thus, σ : S ↷ G c is an automorphic S-action, and exactness of σ : S ↷ G is equivalent to saying that there are no S-invariant subgroups H ⊆ G such that the associated S-action S ↷ H is automorphic.Remark 4.13.Our definition of exactness for an algebraic action is a vast generalization of the notion of exactness for a single endomorphism given by Rohlin in [65].
When σ : S ↷ G is an algebraic dynamical system in the sense of [7], then it is exact if and only if s∈S σ s G = {e}.This stronger condition is called "minimal" in [68].Our condition (M1) is automatically satisfied for the actions in [68,7] (see § 7.2 below), which could explain their choice of terminology (cf.[68, Remark 1.7]).
For F ∈ F, put ∩F := C∈F C, and for H ⊆ G, let core G (H) := g∈G gHg −1 .Our main result on topological freeness reads as follows: Theorem 4.14.Suppose that X ⊆ ∂ E is a closed invariant subspace, and let F = F(X).
(i) If the groupoid I ⋉ X obtained by restricting G σ to X is topologically free, then for all D ∈ F∈F F, we have F∈F, D∈F core G (∩F) = {e}.(ii) If for all D ∈ F∈F F, we have F∈F, D∈F ∩F = {e}, then I ⋉ X is topologically free.
In particular, if G is Abelian, then G σ is topologically free if and only if σ : S ↷ G is exact.
For the proof, we need some preparations.
Proof.Put D := C ∩ i C i .Since D ⊆ C i for all i and D ⊆ C, each of the cosets gC and g i C i can be written as a non-trivial (possibly infinite) disjoint union of D-cosets.Hence, gC \ i g i C i contains a D-coset, hD say.Now suppose i g i C i ⊊ gC.If χ ∈ ∂ E(hD), i.e., χ(hD) = 1, then χ(gC) = 1 because hD ⊆ gC.If we had χ(g i C i ) = 1 for some i, then we would have χ(hD ∩ g i C i ) = 1; this is impossible since hD and g i C i are disjoint.Thus, ∂ E(hD) ⊆ ∂ E(gC; {g i C i }).□ Lemma 4.16.Let X and F be as in Theorem 4.14.The following are true: We also deduce the following consequence, which is special to our situation and in general only holds for amenable groupoids.As in Remark 4.12, set G c := C∈C C.
"(iii) ⇒ (i)": Assume G c is amenable and G c = C∈C core G (C), and suppose that G c ̸ = {e}.Arguing as in the proof of part (i) of Theorem 4.14, we see that  [37,Remark 7.27]).□ In addition to the application to C*-algebras above, pure infiniteness has implications for topological full groups and homology: It was recently proven in [47] that purely infinite groupoids satisfy Matui's AH Conjecture from [53], and general results for topological full groups of purely infinite groupoids have been established in [27].
We have the finite decomposition B = gC ⊔ hC∈B/C,hC̸ =gC hC, so χ(hC) = 1 for some hC ∈ B/C by Lemma 3.22.Since χ(gC Proof.By Theorem 4.10, there exist a constructible subgroup C ⊆ B and φ ∈ I e such that C ⊆ dom(φ) and φ(C) ⊆ G has finite index.Since the partial isomorphism φ −1 maps φ(C) ∩ B onto and all indices are finite.Thus, where all indices are finite, so that [B : .., g m ∈ B be a complete set of representatives for B/C.Since [G : σ s G] ≥ 2, we can find h 1 , ..., h 2m ∈ B such that the cosets h j σ s C are pairwise disjoint for 1 ≤ j ≤ 2m.Then are compact open bisections.By Lemma 3.22, Using Lemma 3.22, we have are disjoint by our choice of h j 's and are contained in ∂ E(hD) because hh k C ⊆ hD for all k.Now put

Comparison of the groupoid model and the concrete C*-algebra
We now compare the C*-algebras of our groupoid G σ with A σ .Throughout this section, σ : S ↷ G will be a non-automorphic algebraic action with S and G countable.

5.1.
Comparison with the essential groupoid C*-algebra.We shall first describe C * ess (G σ ) using induced representations.For this, we need some preliminary results.We let S e := (G σ ) χe χe , where χ e is the character from Definition 3.26.
Proof.The first claim is easy to see.We have χ h = χ g if and only if χ h (kC) = χ g (kC) for all kC ∈ E × , i.e., h ∈ kC if and only if g ∈ kC for all kC ∈ E × .This in turn is equivalent to having hC = gC for all C ∈ C, i.e., g Let us introduce some notation following [24].For ϕ ∈ I × , let F ϕ := {χ ∈ ∂ E : χ(dom(ϕ)) = 1, ϕ.χ = χ} be the set of fixed characters of ϕ, and we let be the set of trivially fixed characters of ϕ.It is straightforward to see that Proof.By [24, Proposition 3.14], we have Proof.First, we claim that if (χ i ) i is a net in ∂ E that converges to χ k for some k ∈ G, then (χ i ) i does not converge to any point of (G σ ) χ k χ k \ {χ k }.In the terminology from [37, § 7], this says that none of the points in {χ k : k ∈ G} are dangerous.Since t −1 k is a homeomorphism of ∂ E taking χ k to χ e , it suffices to consider the case k = e.Suppose [ϕ, χ e ] ∈ S e \ {χ e }.By Lemma 5.2, there exists h ∈ G c and φ ∈ I e such that ϕ = t h φ.We need to show that (χ i ) i does not converge to [ϕ, χ e ].But this follows immediately from Lemma 5.3.
The essential observation is the following.
Let us now make a couple of observations about the group Še .Lemma 5.2 gives the following: Any semigroup P that can be embedded into a group admits a universal group embedding, i.e., there exists a group G univ , called the universal group of P , and an embedding P → G univ such that for any homomorphism from P to a group H extends (uniquely) to a homomorphism G univ → H (see [13, § 12] and the discussion in [16, § 5.4.1]).Thus, if the universal group of the monoid {[σ s , χ e ] : s ∈ S} is amenable, then Še is amenable.If S can be embedded into a group and the universal group of S is amenable, then Še is amenable.
The maximal group image of I e is defined to be the quotient of I e by the congruence ϕ ∼ ψ if there exists if and only if ϕ ∼ ψ, we see that the group Še is canonically isomorphic to the maximal group image of I e .
If λ e Se = λ Se , then (ii) implies λ Se/ Še ⪯ λ Se , which in turns implies that Še is amenable.If I is strongly 0-E-unitary, then G σ is a partial transformation groupoid (see [16,Lemma 5.5 5 is an isomorphism.
We will conclude this section by explaining how A σ is, in many cases, an exotic groupoid C*-algebra.Lemma 5.15.If I is strongly 0-E-unitary and ρ : In fact, this map is injective (see, e.g., [64,Exercise 3.3.6]).It is easy to see that this embedding sends the canonical unitary u g in C(∂ E) ⋊ G corresponding to g ∈ G to v tg .Moreover, the canonical *-homomorphism C * (G) → C(∂ E) ⋊ G is injective because there is a G-invariant probability measure on ∂ E. Now the composition of these canonical embeddings with ρ coincides with the left regular representation λ G of C * (G).Thus, if ρ is an isomorphism, then λ G must also be an isomorphism, in which case G is amenable.□ Corollary 5.16.Suppose σ : S ↷ G is exact and I is strongly 0-E-unitary.If both G and Še are non-amenable, then A σ is an exotic groupoid C*-algebra, in the sense that it sits properly between the full and reduced C*-algebra of G σ .
Proof.Our assumptions imply that G σ is Hausdorff.Hence Proposition 5.5 produces the canonical projection map . ., g n ∈ G and s 1 , . . ., s n ∈ S.Here we identify G ⋊ S with its canonical copy inside I l .If S is left reversible (i.e., sS ∩ tS ̸ = ∅ for all s, t ∈ S), then ∅ / ∈ J S by [16,Lemma 5.6.43], and the projection onto the G-component defines an inverse semigroup homomorphism I l → I, Φ → Φ G .It is straightforward to see that this map is surjective.By restricting the map I l → I, we obtain a surjective semilattice homomorphism E → E and hence a continuous embedding E → E, χ → χ.
For the remainder of this subsection, we assume that S is left reversible.Lemma 6.1.The map E → E, χ → χ restricts to a bijection E max ∼ = E max .
Proof.Recall that χ ∈ E is maximal if and only if whenever χ(gC) = 0 for some gC ∈ E, there exists hD ∈ E with χ(hD) = 1 and gC ∩ hD = ∅.Moreover, every element of E is of the form gC × X for some gC ∈ E and X ∈ J S .Since S is left reversible, we have (gC × X) ∩ (hD × Y ) = ∅ if and only if gC ∩ hD = ∅.Now suppose that χ ∈ E max .Then χ(gC ×X) = 1 if and only if χ(gC) = 1.Assume that χ(gC ×X) = 0. Then χ(gC) = 0. Hence there exists hD ∈ E with χ(hD) = 1 and gC ∩ hD = ∅.It follows that, for every Y such that hD × Y ∈ E, we have χ(hD × Y ) = 1.At the same time, (gC × X) ∩ (hD × Y ) = ∅.This shows that χ is maximal.Now take ω ∈ E max .Define χ ∈ E by χ(gC) = 1 if ω(gC × X) = 1 for some X ∈ J S .Then χ ∈ E max .We claim that χ = ω.Indeed, χ(gC) = 1 if and only if ω(gC × X) = 1 for some X if and only if ω(gC × Z) = 1 for all Z such that gC × Z ∈ E. The last equivalence follows from maximality of ω. □ Proof."⇒" is clear.For "⇐", take Φ and assume that Φ G ∈ E, say Φ G = id gC and χ(gC) = 1, so that [Φ G , χ] ∈ (G σ ) (0) .Thus if I l ⋉ ∂ E ↠ G σ is an isomorphism, we deduce [Φ, χ] = χ and thus Φ| gC×X = id gC×X for some X ∈ J S .Therefore, Φ = Φ G × Φ S with Φ G ∈ E and Φ S | X = id X .As I l (S) is E-unitary, the latter implies that Φ S ∈ E(S), and hence Φ ∈ E, as desired.□ Proof.For all s, t ∈ S, we have σ s G ∩ σ t G ̸ = ∅, so that Λ σs Λ * σs Λ σt Λ * σt ̸ = 0 and v σs v * σs v σt v * σt ̸ = 0. Thus, left reversibility of S is necessary for existence in both cases.Now assume S is left reversible.To prove (ii), it suffices to show that the representation I l → A σ , Φ → Λ Φ G is a tight representation of I l in the sense of [21]; since the restriction to E is unital, it suffices by [23,Corollary 4.3] to prove that this representation is cover-to-join in the sense of [23, § 3].Let gC × X ∈ E × and suppose c ⊆ E is a finite cover of gC × X.Then for every hD × Y ⊆ gC × X, there exists kB × Z ∈ c such that (kB × Z) ∩ (hD × Y ) ̸ = ∅.Let c G := {kB : kB × Z ∈ c for some Z ∈ J S }.It is easy to see that c G is a (finite) cover of gC.We have where the last equality uses that ϕ → Λ ϕ is a cover-to-join representation of I in A σ (see the proof of Proposition 3.30).
The proof of (iii) is essentially the same, using that I l → C * r (G σ ), Φ → v Φ G is a tight representation.□ Remark 6.5.A special case of the equivalence of (i)⇔(ii) in Proposition 6.4 was observed in [5, Proposition 4.3] using different methods.
We lastly compare C * ess (G σ ) and C * ess (I l ⋉ ∂ E).For this, we also assume that S and G are countable, so that we can use results from [57]. ).It remains to check that the unitary ℓ 2 ((I l ⋉ ∂ E) χe /N ) ∼ = ℓ 2 ((G σ ) χe ) associated with the above bijection implements a unitary equivalence between θ and Ind λ Te/N .This is similar to the proof of Proposition 5.6.□ The following is analogous to Proposition 5.12 and Corollary 6.7 Corollary 6.10.Consider the following statements: (i) N is amenable, (ii) λ Te/N ⪯ λ e Te , (iii) θ ⪯ Ind λ Te , i.e., there is a *-homomorphism θ ess : We always have (i) ⇒ (ii) and (i) Te = λ Te (e.g., if P embeds into a group), then (ii) ⇒ (i).
Recall that we are assuming that S is left reversible.Lemma 6.12.Suppose θ ess in Corollary 6.10 (iii) exists (for instance if N is amenable) and that ). □ Remark 6.13.If I l is 0-E-unitary, then our condition above for non-injectivity of θ ess is satisfied whenever there exists Φ / ∈ E with Φ G = id (i.e., Φ G ∈ E), i.e., I l ↠ I is not injective.In other words, if I l is 0-E-unitary, and if θ ess is an isomorphism, then the map I l ↠ I from above must be an isomorphism.In particular, again if I l is 0-E-unitary, then θ ess is not injective whenever I is not 0-E-unitary.Proposition 6.14.Suppose θ ess in Corollary 6.10 (iii) exists (for instance if N is amenable).Then θ ess is an isomorphism if and only if N is trivial.
Proof.If N is trivial, then Ind λ Te/N = Ind λ Te , and the result follows from Propositions 6.8 and 6.9.If N is non-trivial, then θ ess is not injective because for any non-trivial [Φ, χe ] ∈ N , Φ satisfies the condition in Lemma 6.12.□ Remark 7.8.For a non-automorphic algebraic action σ : S ↷ Z r , property ID for the dual action in the sense of [3,55] implies exactness for σ.
Example 7.9 (Actions on infinite rank groups that satisfy (FI)).Fix a left cancellative semigroup S. Suppose J is a non-empty set and that for each j ∈ J we have an algebraic action σ j : S ↷ G j satisfying (FI).Then the diagonal action δ : S ↷ i∈J G j given by (δ s (g j )) j = σ j (g j ) satisfies (FI) if and only if for every s ∈ S, σ j s ∈ End (G j ) is invertible for all but finitely many j.For instance, if O K is the ring of integers in a number field K, then the diagonal action FI), where P K is the set of non-zero prime ideals of O K and O K,[p] is the localization of O K at the p ∈ P K .
Example 7.10 (Algebraic actions from self-similar actions of groups).We briefly explain how to obtain examples of algebraic actions from self-similar group actions.We refer the reader to [56] and [40] for background on self-similar actions.Let d ∈ Z >1 and X := {0, ..., d−1}.Let X * denote the free monoid on X, and for each n ∈ N, let X n ⊆ X * be the set of words of length n.Suppose G ↷ X * is a faithful self-similar action of a non-trivial group G on X * as in [40, § 3] (cf.[56]).For each µ ∈ X * , let G µ ⊆ G be the stabilizer subgroup of µ, and let ϕ µ : G µ → G be the homomorphism ϕ µ (g) := g| µ , where g| µ is the section of g at µ. Assume that ϕ x is an isomorphism for all x ∈ X.By [40,Lemma 3.10], this is equivalent to assuming ϕ µ is an isomorphism for all µ ∈ X * .For each µ ∈ X * , put σ . Thus, our groupoid G σ is topologically free, minimal, and purely infinite.
Example 7.11 (Ring C*-algebras of non-commutative rings).Let R be a unital (not necessary commutative) ring with 1 ̸ = 0, and let R × be the set of left regular elements of R, i.e., R × := {a ∈ R : ax = ay implies x = y for all x, y ∈ R}.Then R × acts on the additive group of R by injective endomorphisms.Since R is unital, the algebraic action R × ↷ R is faithful.The concrete C*-algebra associated with R × ↷ R is called the reduced ring C*-algebra of R (see [41]) and is denoted by A r [R].Assume that the additive group of R is torsion-free and of finite rank.Examples of such rings include integral group rings of finite groups and R n or M n (R), where R is an order in a central simple algebra over an algebraic number field.Assume R × ↷ R is exact (this occurs if and only if there is no group embedding of Q into the additive group of R, e.g., if the additive group of R is isomorphic to Z d for some d ∈ Z >0 ).It is straightforward to check that R × ↷ R satisfies (FI) and that (Q⊗ Z R) * ↷ Q⊗ Z R is a globalization for R × ↷ R that satisfies (JF).By Corollary 7.4, Corollary 5.14 and Theorem 4.19 imply that the following are equivalent:  This condition means that the family of principal constructible subgroups is co-final in C.
Proposition 7.12.If S is left reversible, then σ : S ↷ G satisfies (PC).  .Let m A (u) = r i=1 f i (u) k i be the factorization of m A (u) into powers of irreducible elements, which are defined up to multiplication by ±1 (here, we are using that Z[u] is a unique factorization domain with unit group {±1}).
By Proposition 7.20 applied to S = u N ∼ = N, the canonical N-action N ↷ Z[u]/(f i (u)) generated by g(u) + (f i (u)) → ug(u) + (f i (u)) is exact if and only if u + (f i (u)) is a non-unit, which is equivalent to f i (0) / ∈ {±1}.Thus, in order to deduce Krzyżewski's characterization of exactness from Proposition 7.25 it suffices to prove the following lemma.Proof.If p ∈ Asc Z[u] (Z n ), then Z[u]/p embeds as a Z[u]-submodule of Z n ; in particular, Z[u]/p is torsion-free as an additive group, so that p ∩ Z = (0).Hence, taking localizations with respect to Z × and applying [19, Theorem 3.1(c)] gives us (8) Asc where p Q denotes the prime ideal of Q[u] generated by p.Let q ∈ Asc Q[u] (Q n ).Since Q[u] is a principal ideal domain, we can write q = (p(u)) Q for some irreducible polynomial p(u) ∈ Z[u].Here, we write (p(u)) Q for the ideal of Q[u] generated by p(u).Since (p(u) , so that (p(u)) Q = (f i (u)) Q for some 1 ≤ i ≤ r.Thus, we have Fix 1 ≤ i ≤ r, and let x ∈ (m A (u)/f i (u)).Q n \ {0} (such an x exists by definition of m A ). Then f i (u).x = 0, so (f i (u)) Q ⊆ Ann Q[u] (x).Since 1.x ̸ = 0, Ann Q[u] (x) is a proper ideal of Q[u]; since (f i (u)) Q is a prime ideal and Q[u] has Krull dimension 1, we must have (f i (u) Therefore, using (8), we have {p   If, in addition, S is right LCM (i.e., J × S = {sS: s ∈ S}) and S * = {1}, then our K-theory formula simplifies to Note that if S is right LCM, then the condition that #(X \ Y ) = ∞ for all X, Y ∈ J × S with Y ⊊ X is equivalent to #(S \ sS) = ∞ for all s ∈ S \ S * .Let us present two example classes where the latter condition holds.
Example 7.42.Assume that S is right LCM and s ∈ S \ S * satisfies #(S \ sS) < ∞, say S \ sS = {r j }.Further suppose that S is right Noetherian, i.e., we cannot find an infinite chain of the form . . .⊋ Ss 3 ⊋ Ss 2 ⊋ Ss 1 for s 1 , s 2 , s 3 , . . .∈ S. Then every element of S is of the form s n r j for some non-negative integer n and r j ∈ S \sS.Indeed, since S is right Noetherian, for every x ∈ S there exists a maximal non-negative integer n such that x = s n y with y / ∈ sS, so that y ∈ {r j }.In particular, if S is Abelian and cancellative, then this would imply that the enveloping group S = S −1 S is virtually Abelian.
Example 7.43.Now suppose that S is given by generators and relations, i.e., S = ⟨Γ|R⟩ + as in [48, § 2.1.1],with the same standing assumptions as in [48, § 2.1.2].Assume that for every generator γ ∈ Γ there exists an infinite word w in Γ not starting with γ such that no relator appears as a finite subword of w (in particular, this implies that #Γ > 1 and no generator is a relator).If w l denotes the finite subword of w consisting of the first l letters, then our assumptions imply that w l / ∈ γS as well as w l ̸ = w m whenever l ̸ = m, so that #(S \ γS) = ∞.It then follows that #(S \ sS) = ∞ for all s ∈ S \ S * .
Let us now present isomorphism results for two classes of full shifts.
Corollary 7.44.Assume that S i , i = 1, 2, are two non-trivial, countable, left reversible monoids which are cancellative, right LCM, satisfy S * i = {1} as well as #(S i \ s i S i ) = ∞ for all 1 ̸ = s i ∈ S i , and that their enveloping groups S i are amenable.Let Σ i , i = 1, 2, be any two non-trivial finite Abelian groups.Consider the full shifts σ i : S i ↷ S i Σ i .Then we have Proof.This follows from Corollary 7.34, (11) and the Kirchberg-Phillips classification theorem [35,61], together with the observation that for any infinite, countable monoid S and any non-trivial, finite, Abelian group Σ, C * λ ( S Σ) is isomorphic to the C*-algebra of continuous functions on the Cantor space, hence independent of S and Σ. □ Corollary 7.45.Assume that S i , i = 1, 2, are two non-trivial, countable, left reversible monoids which are cancellative, right LCM, satisfy S * i = {1}, and that their enveloping groups S i are amenable.Let Σ be an arbitrary infinite, amenable group.Consider the full shifts σ i : S i ↷ S i Σ.Then we have Proof.Apply Corollary 7.34, (11) and the Kirchberg-Phillips classification theorem [35,61] B and s| B are injective on B. Given an open bisection U ⊆ G, we let C c (U ) denote the set of continuous compactly supported complex-valued functions on U .Extension-by-zero gives an embedding C c (U ) ⊆ ℓ ∞ (G), and we let C(G) be the linear subspace of ℓ ∞ (G) spanned by the subspaces C c (U ) as U runs through the open bisections of G. Then C(G) carries the natural structure of a *-algebra (see, e.g., [21, § 3]).The (full) C*-algebra of G, which we denote by C * (G), is the enveloping C*-algebra of C(G).For each unit x ∈ G (0) , there is a representation π with respect to the norm ||f || r := sup x∈G (0) ||π x (f )||.We shall view each π x as a representation of C * r (G), and let π r : C * (G) → C * r (G) be the projection map.

3. 1 .
The concrete C*-algebra associated with an algebraic action.Each algebraic action σ : S ↷ G naturally gives rise to a concrete C*-algebra acting on ℓ 2 (G): Define an isometric representation κ σ : S → Isom(ℓ 2 (G)) by κ σ (s)δ h = δ σs(h) Moreover, set F(χ) := F(Π(χ)) := {C ∈ C: Π(χ)(C) = 1}.Lemma 3.24.(i) For every χ ∈ E, F(χ) is a filter on C. (ii) Π is a surjective map from E onto the space of characters of C.More precisely, for each filter F on C there exists a character χ(F) ∈ E uniquely determined by F(χ(F)) = F.Moreover, for every filter F on C, χ ∈ E lies in Π −1 (F) if and only if χ(g ′ C ′ ) = 0 for all C ′ ∈ C with C ′ / ∈ F, and for all C ∈ F, there exists g C ∈ G such that χ(g C C) = 1, and (g C ) satisfies the compatibility condition that g C ∈ g D D for all C, D ∈ F C with C ⊆ D. (iii) Given χ ∈ E, we have χ ∈ ∂ E if and only if F(χ) is a finitely hereditary filter.(iv) The maximal filter on C is given by

Definition 3 . 26 .
Given k ∈ G, we denote by χ k the character of E which satisfies χ(kC) = 1 for all C ∈ C.

Corollary 4 . 7 .
The map in Theorem 4.4 implements a one-to-one correspondence between closed, G-invariant subspaces of ∂ E and ⊆-closed subsets of F C .

Definition 4 . 11 .
The algebraic action σ : S ↷ G is called exact if C∈C C = {e}.

Remark 4 . 24 .
Condition (M1) implies that {e} / ∈ C. If {e} ∈ C, then A σ ∼ = C *ess (G σ ) (see Corollary 5.7 below), and A σ contains the compact operators, so C * ess (G σ ) cannot be purely infinite in this case.Before proceeding to the proof of Theorem 4.22, we need two lemmas.

Lemma 4 . 25 .
Let B ∈ C, and let {k i B i } ⊆ E × be a finite (possibly empty) collection.Then for all gC ∈ E × with gC ⊆ B and 1 < [B : C] < ∞, we have by our choice of h j 's.Thus, ∂ E(B) is properly infinite.Now let us suppose there exists a constructible subgroupC ′ ⊆ B such that [B : C ′ ] = ∞.By Lemma 4.15, there exists a constructible coset hD ⊆ B \ i k i B i , so that ∂ E(hD) ⊆ ∂ E(B; {k i B i }).By replacing D with D ∩ C ′ , we may assume that D ⊆ C ′ , so that [B : D] = ∞.If [D : D ′ ] < ∞ for all constructible subgroups D ′ ⊆ D,then [G : D] < ∞ by Lemma 4.26; since the inclusion map B/D → G/D is injective, this would then imply that [B : D] < ∞, a contradiction.Thus, there exists a constructible subgroup D ′ ⊆ D such that [D : D ′ ] = ∞.By Theorem 4.10, there exist a constructible subgroup C ⊆ D ′ and φ ∈ I e with C ⊆ dom(φ) such that φ(C) ⊆ B has finite index.Let g 1 , ..., g m be a complete set of representatives for B/φ(C).Choose h 1 , ..., h 2m ∈ D such that the cosets h i C are pairwise disjoint for 1 ≤ j ≤ 2m (here we are using that [D : C] = ∞).Consider the compact open bisections

Lemma 5 . 2 .
−1 h ∈ C for all C ∈ C. □ We have S e = {[t h φ, χ e ] : h ∈ G c , φ ∈ I e }.Proof.It follows from Lemma 5.1 that [t h φ, χ e ] ∈ S e for all h ∈ G c and φ ∈ I e .Suppose [ϕ, χ e ] ∈ S e .Then ϕ.χ e = χ e , so in particular e ∈ dom(ϕ) which by Proposition 3.12 implies that dom(ϕ) is a constructible subgroup and ϕ = t h φ for some h ∈ G and φ ∈ I e .Since χ e = ϕ.χ e = χ ϕ(e) = χ h , we have h ∈ G c by Lemma 5.1.□ χ e , we have t h φ| D ̸ = id D for all D ∈ C. Suppose there exists gC ∈ E × \ C with gC ⊆ fix(t h φ).Then g ∈ dom(φ) = dom(t h φ), and since φ ∈ I e , dom(φ) is a (constructible) subgroup, so we then get C ⊆ dom(φ).Since φ is a group homomorphism on its domain (see Remark 3.9), we have gc = t h φ(gc) = hφ(g)φ(c) = hgφ(c) for every c ∈ C. Taking c = e gives g = hg, so that h = e.But now we have gc = gφ(c) for all c ∈ C, which implies φ| C = id C , a contradiction.Hence, fix(t h φ) contains no constructible cosets, so that T F t h φ = ∅.□Let λSe be the left regular representation of the group C*-algebra C * (S e ), and let Ind λ Se be the representation of C * (G σ ) on ℓ 2 ((G σ ) χe ) induced from λ Se .Inspired by a recent result in the setting of semigroup C*-algebras from [57], we now prove the following: Proposition 5.4 (cf.[57, Proposition 2.5]).We have C * ess (G σ ) = (Ind λ Se )(C * (G)).

Corollary 5 . 10 .Remark 5 . 11 .
We have S e = Še if and only if σ : S ↷ G is exact.If we have a globalization σ : S ↷ G that satisfies (JF), so that G σ ∼ = (G ⋊ S ) ⋉ ∂ E by Remark 3.32, then S e ∼ = G c ⋊ ⟨S⟩ and Še ∼ = ⟨S⟩ by Lemma 5.2, where ⟨S⟩ is the subgroup of S generated by S.
to characterize ρ ⪯ Ind λ Se .Let C * e (S e ) denote the completion of the complex group algebra CS e with respect to the norm || • || e from [12, Definition 2.1], i.e., || • || e is given by ||f || e := sup{||π(f )|| : π a representation of C * (S e ) such that Ind π ≺ λ Se } for all f ∈ CS e .Denote by λ e Se the canonical projection C * (S e ) → C * e (S e ).The following is a consequence of Proposition 5.6.

6 . 6 . 1 .
then ρ is not injective by Lemma 5.15.Our claim follows.□ Example 7.35 contains a concrete example class where the hypotheses in Corollary 5.16 are satisfied.Note that, by Proposition 3.16, I is strongly 0-E-unitary if σ : S ↷ G has a globalization σ : S ↷ G which satisfies (JF).Comparisons with the boundary quotient of the semigroup C*-algebra It is natural to compare our C*-algebras C * ess (G σ ), C * r (G σ ), and A σ to the boundary quotient of the semigroup C*-algebra C * λ (P ), where P = G ⋊ S. For background on semigroup C*-algebras and their boundary quotients, see [46, § 2] and [16, Chapter 5].Comparison of groupoids.Let I l be the left inverse hull of P , E the semilattice of idempotents of I l , and I l ⋉ ∂ E the associated boundary groupoid, so that C * r (I l ⋉ ∂ E) is the boundary quotient of the semigroup C*-algebra C * λ (P ).Let J S denote the semilattice of constructible right ideals of S. A straightforward computation shows

Corollary 6 . 2 .Lemma 6 . 3 .
The map E → E, χ → χ restricts to a homeomorphism ∂ E ∼ = ∂ E. Now we see that there is a canonical surjection I l ⋉ ∂ E ↠ G σ given by [Φ, χ] → [Φ G , χ].If the left inverse hull I l (S) of S is E-unitary, then the surjection I l ↠ I, Φ → Φ G , is an isomorphism if and only if the surjection

6. 2 .Proposition 6 . 4 .
Comparison of C*-algebras.Let us now compare the C*-algebras.For Φ ∈ I l , let w Φ denote the corresponding partial isometry in C(I l ⋉ ∂ E).The following are equivalent:(i) S is left reversible;(ii) there exists a *-homomorphism ϑ : C

7. 2 . 1 .
General results for actions satisfying (PC).We shall call S-constructible subgroups of the form σ s G principal constructible subgroups.This terminology comes from the ring-theoretic examples where the principal constructible subgroups are principal ideals of the ring.Consider the following condition on σ : S ↷ G:(PC)For every C ∈ C, there exists s ∈ S such that σ s G ⊆ C.

Proof.
Let C ∈ C. Since σ : S ↷ G satisfies (PC), there exists s ∈ S such that σ s G ⊆ C. Now G = σ −1 s C, so (M2) holds.□ Corollary 7.15.Assume G is Abelian, and S is cancellative and right reversible (i.e., left Ore).If σ : S ↷ G is faithful and satisfies (PC), then σ : ⟨S⟩ ↷ S −1 G satisfies (JF).Proof.For convenience, let us write G additively.Since S is left Ore, by (iii) in Example 3.17, it suffices to show that C ⊆ ker (σ s − σ t ) =⇒ s = t for all C ∈ C and s ∈ S. Suppose we have C ∈ C and s, t ∈ S with C ⊆ ker (σ s − σ t ).By assumption, S ↷ G satisfies (PC), so there exists r ∈ S such that σ r G ⊆ C, so we have σ s σ r (g) = σ t σ r (g) for all g ∈ G.By faithfulness, it follows that sr = tr, and hence s = t by right cancellation.□ Lemma 7.16.Assume that σ : S ↷ G has a globalization σ : S ↷ G and that (PC) holds.Then (JF) is satisfied if and only if σg | G = id implies g = 1 for all g ∈ G ⋊ S .Proof.Assume σg | G = id implies g = 1 for all g ∈ G ⋊ S .Suppose σg | C = id C for some C ∈ C. Then by (PC), we can find s ∈ S with σ s G ⊆ C. Now we have that σg • σs and σs agree on G. Therefore our assumption implies gs = s and thus g = 1.□ Since A has integer entries, m A (u) ∈ Z[u]
is a monic polynomial, then (f (u)) Q ∩ Z[u] = f Z[u] by Gauss's lemma.□ Example 7.28 (Algebraic N d -actions).Fix d ∈ Z >0 , and let R + d := Z[u 1 , ..., u d ] be the polynomial ring with integer coefficients in the d commuting variables u 1 , ..., u d .For n = (n 1 , ..., n d ) ∈ N d , we let u n := u n 1 1 • • • u n d d .Given f ∈ R + d , we can write f = n∈N d f n u n , where f n ∈ Z is zero for all but finitely many n.Given any algebraic N d -action N d ↷ M , where M is an Abelian group, M naturally becomes a module over R + d via f.x := n f n u n .x.Note that R + d is Noetherian (see, e.g., [2, Theorem 7.5]).Proposition 7.29.Let M be a finitely generated module over Z[u 1 , ..., u d ], and assume that u i / ∈ p for every 1 ≤ i ≤ d and every p ∈ Asc(M ), so that we get an algebraic N d -action N d ↷ M as in Example 7.28.Then (i) N d ↷ M is faithful if and only if there exists p ∈ Asc(M ) such that (9){u n − u m : n, m ∈ N d } ∩ p = {0}; (ii) Let π j : R + d ↠ Z[u 1 , . . ., u j−1 , u j+1 , . . ., u d ] be the canonical projection.N d ↷ M is exact if and only if for all p ∈ Asc(M ) there exists 1 ≤ j ≤ d such that 1 / ∈ π j (p).In particular, N d ↷ M is exact if we can find f 1 , ..., f m ∈ Z[u 1 , ..., u d ] and z ∈ Z d with z j = 0 for some 1 ≤ j ≤ d such that p = (f 1 , ..., f m ) and gcd(f 1 (z), ..., f m (z)) ̸ = 1.Proof.(i): By Lemma 7.24, N d ↷ M is faithful if and only if there exists a prime p ∈ Asc(M ) such that N d ↷ R + d /p is faithful, which happens if and only if {u n − u m : n, m ∈ N d } ∩ p = {0}.(ii): By Example 7.23 and Proposition 7.25, N d ↷ M is exact if and only if the image of the canonical map ⟨u 1 , ..., u d ⟩ + → R + d /p contains a non-unit for every p ∈ Asc(M ).
we see that c is a cover of the constructible coset gC if and only if gC = i h i D i .
□Let us now give a characterization of tight characters in our situation.Lemma 3.22.Let χ ∈ E. Then χ lies in ∂ E if and only if for all gC ∈ E × with χ(gC) = 1 and all [23,nce ρ is unital,[23, Corollary 4.3]implies that Λ is tight if and only if it is cover-to-join in the sense of[23,  § 3], i.e., for every gC ∈ E × , we have Now suppose c = {g i C i : i ∈ F } is a finite cover of gC.By Lemma 3.21, we have gC = i∈F g i C i .In D σ , we have hD∈cΛ(hD)for all finite covers c of gC.
Assume (without loss of generality) that S is generated by S. If G σ is amenable, then S is amenable.The converse holds if G is amenable.Proof.As explained in § 3.5, our assumptions mean that we have the identification G σ ∼ = (G ⋊S )⋉∂ E. Our assumptions also give S ⊆ (G σ ) χe χe , so amenability of S follows from amenability of G σ by[1, Proposition 5.1.1].If G and S are amenable, then G ⋊ S is amenable, so that (G ⋊ S ) ⋉ ∂ E is amenable.□Remark4.20.In general, C * r (G σ ) is nuclear if and only if G σ is amenable (see [1]).Assume we are in the setting of Theorem 4.19, so that G σ ∼ = (G ⋊S )⋉∂ E. If the group G ⋊S is exact, then C * r (G σ ) is nuclear if and only if the canonical map C * (G σ ) → C * r (G σ ) is an isomorphism by [11, Theorem 4.12].4.6.Pure infiniteness.We now turn to pure infiniteness following [54, § 4].Let G be an ample étale groupoid with compact unit space.A subset X ⊆ G (0) is said to be properly infinite if there exist compact open bisections U, V ⊆ G such that s(U ) = s(V ) = X and r(U ) ⊔ r(V ) ⊆ X.For many classes of étale groupoids, one has a dichotomy (see, e.g., [4, 62, 50]): Either the groupoid is purely infinite, or the groupoid admits an invariant measure.Indeed, we have the following open question: By Theorem 4.14, G σ is topologically free, and by Theorem 4.10, G σ is minimal.The groupoid G σ is purely infinite by Theorem 4.22, which implies G σ is locally contacting.Now the result follows from [37, Theorem 7.26] (see χ]}.Remark 4.12 implies that (χ, [G c , χ]): χ ∈ ∂ E is an essentially confined amenable section of isotropy groups of G σ , in the sense of[32, Definition 7.1].Hence [32, Theorem 7.2] implies that C(∂ E) ⊆ C * ess (G σ ) does not have the ideal intersection property.□ 4.5.Amenability.We now consider amenability for our groupoids.Theorem 4.19.Suppose σ : S ↷ A has a globalization σ : S ↷ G and that (JF) is satisfied.Question 4.21.Let G be a second countable, topologically free, minimal, ample étale groupoid with compact unit space.If there is no invariant measure on G (0) , then must G be purely infinite?For our groupoid G σ , if it is minimal, then the action σ : S ↷ G must be non-automorphic, from which it is easy to see that ∂ E is properly infinite, so that ∂ E has no invariant measures.Thus, Theorem 4.22 below answers Question 4.21 in the affirmative for the class of groupoids arising from algebraic actions.Theorem 4.22.If G σ is minimal, then G σ is purely infinite.Combining this with our results above and general results for groupoid C*-algebras, we obtain: Corollary 4.23.Suppose σ : S ↷ G is a non-automorphic algebraic action.If σ : S ↷ G is exact and satisfies one of the conditions in Theorem 4.10, then C * ess (G σ ) is simple and purely infinite.Proof.
[43,FI) holds.□Wearenowreadyfor the proof of our theorem.Proof of Theorem 4.22.By[43, Lemma 4.1], it suffices to prove that every non-empty basic open set ∂ E(kB; {k i B i }) is properly infinite.Suppose we are given ∂ E(kB; {k i B i }) ̸ = ∅, where kB ∈ E × and {k i B i } is a finite (possibly empty) collection of constructible cosets.By conjugating by the homeomorphism t k if necessary, we may assume k = e.We may also assume k i B i ⊊ B for all i.
First, let us suppose [B : C] < ∞ for all C ∈ C with C ⊆ B. By Lemma 4.25, we may assume {k i B i } = ∅ in this case.By Lemma 4.26, [G : B] < ∞.Since σ : S ↷ G is non-automorphic, there exists s ∈ S such that σ s .22.], for instance), so || • || e and || • || r coincide by [12, Corollary 4.15], i.e., λ e Se = λ Se .□ Remark 5.13.In general, it is not clear when || • || e = || • || r on CS e , i.e., when λ e Se = λ Se .Corollary 5.14.Assume that σ : S ↷ G has a globalization σ : S ↷ G which satisfies (JF), σ is exact and one of the conditions in Theorem 4.10 is satisfied.Then ⟨S⟩ is amenable if and only if A σ is simple if and only if the map . Examples 7.1.Algebraic actions with the finite index property.−1 t C) = σ t G/((σ t G) ∩ C), and the latter is finite because we have an embedding σ t G/((σ t G) ∩ C) → G/C.□ Corollary 7.3.If σ : S ↷ G satisfies (FI), then Ḡ := lim ← −C∈C G/C is compact.Moreover, every character in ∂ E is maximal, and Ḡ coincides with ∂ E (cf.Lemma 3.24).If σ : S ↷ G satisfies (FI), then G σ is minimal.Assume G has the property that g m = h m implies h = g for all g, h ∈ G and all m ∈ Z >0 and that σ : S ↷ A satisfies (FI).Then I is 0-E-unitary.If σ : S ↷ G admits a globalization σ : S ↷ G , then it satisfies (JF).Proof.By Corollary 3.13, it suffices to prove that I e is E-unitary.Suppose φ ∈ I e is such that φ| C = id C for some C ∈ C. By Proposition 7.2, for every g ∈ G there exists m ∈ Z >0 such that g m ∈ C. Now we have φ(g) m = φ(g m ) = g m , which implies φ(g) = g for all g ∈ G by our assumption.The proof that (JF) is satisfied is similar.□ Note that if G is Abelian and torsion free, then g m = h m implies h = g for all g, h ∈ G and all m ∈ Z >0 .Moreover, if I is 0-E-unitary, then G σ is Hausdorff by [24, Corollary 3.17].
Definition 7.1.We say that σ : S ↷ G has the finite index property if (FI) #(G/σ s G) < ∞ for all s ∈ S. Proposition 7.2.If σ : S ↷ G satisfies (FI), then every member of C is a finite index subgroup of G.Proof.We proceed by induction.The induction start is provided by (FI).For the induction step, suppose C ∈ C with #(G/C) < ∞.Now let s, t ∈ S. Since σ s G/σ s C = σ s (G/C), we see that [σ s G : σ s C] is finite by the induction hypothesis.Hence, since [G : σ s G] is also finite by the induction hypothesis, [G : σ s C] is finite (see, e.g., [29, Chapter I, Theorem 4.5]).Moreover, we have σ t (G/σ Example 7.6 (Algebraic actions on tori and solenoids).Let G be a torsion-free Abelian group of finite rank r ∈ Z >0 , so that we can view G as a subgroup of G := Q r .Note that these assumptions on G are equivalent to the dual group G being a solenoid.Given an algebraic action σ : S ↷ G, every σ s extends naturally to an automorphism σs of G , so that we obtain a natural globalization by considering the action of the subgroup S of Aut(G ) generated by σs , s ∈ S.Moreover, (JF) is satisfied by Proposition 7.5, so that G σ ∼ = (G ⋊ S ) ⋉ ∂ E. In particular, our groupoid is Hausdorff.By [26, Exercise 92.5], (FI) is satisfied, so G σ is minimal by Corollary 7.4 and hence purely infinite by Theorem 4.22.Remark 7.7.Example 7.6 provides many algebraic actions which have globalizations even though the acting semigroup is not left Ore.For instance, it is not difficult to find faithful non-automorphic (even exact) actions of free monoids on tori.Such actions are necessarily very far from respecting the order in the sense of [6, Definition 8.1].
] is simple.If the above equivalent conditions are satisfied, then A r [R] is a UCT Kirchberg algebra by Corollary 4.23.7.2.Algebraic actions by left reversible monoids.Recall that the monoid S is said to be left reversible if sS ∩ tS ̸ = ∅ for all s, t ∈ S. Right reversibility is defined analogously.We shall now demonstrate that our conditions from § 4 are especially easy to check for actions by left reversible monoids.
If G is Abelian, then s∈S σ s G = {e} if and only if s∈S fix(σ s ) is dense in G.
The latter holds if and only if there exists 1 ≤ j ≤ d such that u e j + p is not a unit in R + d /p, where {e j } denotes the canonical generators of N d .Now u e j + p is not a unit in R + Proof.If σ : S ↷ G satisfies (II), then every character on E is tight by Lemma 3.22.Our claims follow immediately.□ Example 7.41.Assume S embeds into a countable group S .Let S ↷ S Σ be the full S-shift over a non-trivial group Σ such that ( S Σ) ⋊ S satisfies the Baum-Connes conjecture with coefficients.If either #Σ = ∞ or #(X \ Y ) = ∞ for all X, Y ∈ J × S with Y ⊊ X, then S ↷ S Σ satisfies (II) by Proposition 7.31, so that Theorem 7.38 gives us K * (C * r (G σ )) ∼ = d /p if and only if 1 / ∈ (u e j ) + p if and only if 1 / ∈ π j (p).□ 7.3.Shifts over semigroups.Throughout this section, let S be a left cancellative monoid with S ̸ = S * and Σ any non-trivial group.As before, we denote the identity of Σ by e.Let us recall the definition of shifts over semigroups from Example 2.6.