An algebraic characterization of ample type I groupoids

We give algebraic characterizations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of their Boolean inverse semigroups of compact open bisections. It yields in turn algebraic characterizations of both properties for inverse semigroups in terms of subquotients of their booleanizations.

Theorem B. Let G be a second countable, ample Hausdorff groupoid. Then G is type I if and only if the following two conditions are satisfied.
• No corner of Γ(G) has a non virtually abelian group quotient, and • Γ(G) does not have an infinite, monoidal and 0-simplifying subquotient.
Historically the notation of CCR and type I C * -algebras was motivated by problems in representation theory. Roughly speaking, groups enjoying one of these properties have a well behaved unitary dual. Originally studied in the context of Lie groups and algebraic groups [Har53; Dix57; Ber74; BE20], other classes of non-discrete groups were considered more recently [Cio15;HR19]. The question which discrete groups are CCR and type I was answered conclusively by Thoma [Tho68], characterizing them as the virtually abelian groups. A more direct proof of Thoma's result was obtained by Smith [Smi72] and his original proof was the basis for a Plancherel formula for general discrete groups recently obtained by Bekka [Bek20]. The fundamental nature of CCR and type I C * -algebras also led to study other group like objects such as the aforementioned characterisation of groupoids with these property by Clark [Cla07]. Compared with Thoma's characterisation of discrete type I groups, our Theorems A and B can be considered an analogue for Boolean inverse semigroups. Naturally, the question arises whether a similar characterisation can be obtained for inverse semigroups. The bridge between these structures is provided by the booleanization of an inverse semigroup [LL13;Law20b], which we denote by B(S). In view of a direct algebraic construction of the booleanization we expose in Section 2.1, the next two results give an intrinsic characterisation of CCR and type I inverse semigroups. The booleanization of an inverse semigroup was first introduced in [Law20b]. Our description is more natural from an operator algebraic point of view and equivalent to the original definition of Lawson. This paper contains five sections. After the introduction, we expose necessary preliminaries on (Boolean) inverse semigroups, ample groupoids and noncommutative Stone duality. In Section 3, we study separation properties of the orbit space of a groupoid and obtain algebraic characterisations of groupoids with (T 1 ) and (T 0 ) orbit spaces. In Section 4, we relate isotropy groups of a groupoid to subquotients of the associated Boolean inverse semigroup. The proofs of our main results are collected in Section 5.

Preliminaries
In this section, we recall all notations relevant to our work and introduce some elementary constructions that will be important throughout the text. For inverse semigroups the standard reference is [Law98]. The survey article [Law20a] describes recent advances and the state of the art in inverse semigroup theory. Further, [Law20b] and [Law20a, Sections 3 and 4] provide an introduction to Boolean inverse semigroups. It has to be pointed out that the definition of Boolean inverse semigroups used in the literature has changed over the years, so that care is due when consulting older material. The standard reference for groupoids and their C * -algebras is [Ren80]. For groupoids attached to inverse semigroups and Boolean inverse semigroups, we refer to [Li17; Law12].

Inverse semigroups and Boolean inverse semigroups
In this section we recall the notions of inverse semigroups, Boolean inverse semigroups and the link between them provided by the universal enveloping Boolean inverse semigroup of an inverse semigroup, termed booleanization.
An inverse semigroup S is a semigroup in which for every element s ∈ S there is a unique element s * ∈ S satisfying ss * s = s and s * ss * = s * . The set of idempotents E(S) ⊆ S forms a commutative meet-semilattice, when endowed with the partial order e ≤ f if ef = e. We denote by supp s = s * s and im s = ss * the support and the image of an element s ∈ S, which are idempotents. The partial order on E(S) extends to S by declaring s ≤ t if supp s ≤ supp t and t supp s = s. Given an inverse semigroup S, we denote by S 0 the inverse semigroup with zero obtained by formally adjoining an absorbing idempotent 0. In particular, we will make use of groups with zero. A character on E(S) is a non-zero semilattice homomorphism to {0, 1}. We will denote by E(S) the space of characters on E(S) equipped with the topology of pointwise convergence.
Let S be an inverse semigroup with zero. Two elements s, t ∈ S are called orthogonal, denoted s ⊥ t, if st * = 0 = s * t. A Boolean inverse semigroup is an inverse semigroup with zero, whose semilattice of idempotents is a generalized Boolean algebra such that finite families of pairwise orthogonal elements have joins. Recall that a generalized Boolean algebra can be conveniently described as a Boolean rng. Given two Boolean inverse semigroups B and C, and an inverse semigroup morphism φ : B → C, we say that φ is a morphism of Boolean inverse semigroups if it preserves the joins of orthogonal elements.
To every inverse semigroup S, one associates the enveloping Boolean inverse semigroup or booleanization S ⊆ B(S), which satisfies the universal property that for every Boolean inverse semigroup B and every semigroup homomorphism S → B, there is a unique extension to B(S) such that the following diagram commutes.
The enveloping Boolean inverse semigroup is conveniently described as the left adjoint of the forgetful functor from Boolean inverse semigroups to inverse semigroups with zero [Law20b]. We will use the following concrete description. Consider a semigroup S. The semigroup algebra I(S) = F 2 [E(S)] is a Boolean rng and whose characters (as a rng) are in one to one correspondence with characters of E(S). We consider the following set of formal sums. We consider the equivalence relation given by the following condition. We have i s i e i ∼ j t j f j if and only if e i f j = 0 implies that there is some p ∈ E(S) such that s i p = t j p. One readily checks that the quotient of C(S) by this equivalence relation is a Boolean inverse semigroup. Thanks to the existence of joins of orthogonal families, every map of semigroups with zero into a Boolean inverse semigroup S 0 → B extends uniquely to a map C(S)/∼ → B. By uniqueness of adjoint functors, this shows that C(S)/∼ ∼ = B(S) is the booleanization of S.

Ample groupoids
In this section we fix our notation for groupoids and recall some basic results. We recommend [Ren80] and [Pat99] as resources on the topic.
Given a groupoid G, we denote its set of units by G (0) and the range and source map by r : G → G (0) and d : G → G (0) , respectively. Throughout the text, G will be a topological groupoid meaning it is equipped with a topology making the multiplication and inversion continuous. A bisection of a topological groupoid G is a subset U ⊆ G such that the restrictions d| U and r| U are homeomorphisms onto their images. We call G étale if its topology has a basis consisting of bisections. It is called ample if its topology has a basis consisting of compact open bisections.
If G is a groupoid and A ⊆ G (0) is a set of units, we denote by We denote by Iso(G) the union over all isotropy groups, considered as subsets of G. Then G is effective if the interior of Iso(G) \ G (0) is empty. Given a unit x ∈ G (0) , its orbit is denoted by Gx ⊆ G (0) . We call G minimal if all its orbits are dense.
The set of orbits of a topological groupoid inherits a natural topology. We will be interested in separation properties of this orbit space. First, we observe that the orbit space of a groupoid is a (T 1 )-space if and only if its orbits are closed. An analogue characterisation of groupoids whose orbit space is a (T 0 )-space, is the subject of the Ramsey-Effros-Mackey dichotomy, which we now recall.
Proposition 2.1. Let G be a second countable ample groupoid. Then the following statements are equivalent.
• The orbit space of G is (T 0 ).
• G has a self-accumulating orbit.
• Orbits of G are locally closed.
Proof. The equivalence between the first two items follows from [Ram90, Theorem 2.1, (2) and (4)]. In order to prove the equivalence to the last item, we want to apply [Ram90, Theorem 2.1, (4) and (5)]. To this end, we have to show that the equivalence relation induced by G on X = G (0) is an F σ -subset of X ×X. The map G → X ×X restricted to any compact open bisection of G has a closed image. We conclude with the observation that there are only countably many compact open bisections, since G is second countable.

Noncommutative Stone duality
Classical Stone duality establishes an equivalence of categories between locally compact totally disconnected Hausdorff topological spaces and generalized Boolean algebras. If X is such a space, the generalized Boolean algebra associated with it is CO(X), the algebra of compact open subsets of X. Vice versa, given a generalized Boolean algebra B, its spectrum B is a locally compact totally disconnected Hausdorff topological space. Noncommutative Stone duality generalises this correspondence to an equivalence between ample Hausdorff groupoids and Boolean inverse semigroups. We refer the reader to [Law10;Law12].
Given an ample Hausdorff groupoid G, we denote by Γ(G) the set of compact open bisections of G, which is a Boolean inverse semigroup. Conversely, given a Boolean inverse semigroup B, the set of ultrafilter for the natural order on B forms an ample Hausdorff groupoid G(B). It follows from [Law12, Duality Theorem] that these two operations are dual to each other. See also [Law20a,Theorem 4.4] and [LV19, Theorem 4.2]. We will not need to specify the morphisms of the categories involved in this duality. Invoking [LV19, Theorem 1.2], the Paterson groupoid of an inverse semigroup S can now be identified with G(B(S)).
Noncommutative Stone duality establishes a dictionary between properties of Boolean inverse semigroups and ample Hausdorff groupoids. We recall some of its aspects that will be needed in this piece.

Corners and subgroupoids
Given an ample Hausdorff groupoid G, idempotents in Γ(G) correspond to compact open subsets of G. Given such idempotent p ∈ E(Γ(G)) corresponding to U ⊆ G (0) , the corner pΓ(G)p is naturally isomorphic with Γ(G| U ). Further, there is a one-to-one correspondence between open subgroupoids of G and Boolean inverse subsemigroups B ⊆ Γ(G) assigning the groupoid B ⊆ G to a Boolean inverse semigroup B ⊆ Γ(G).

Morphisms
Noncommutative Stone duality does not cover all morphisms that one would naturally consider in the respective category. This is why it is necessary and useful to note the following two statements. They ensure compatibility of noncommutative Stone duality with restriction maps. The next proposition is a reformulation of [LV19, Proposition 5.10]. See also [Len08].
Proposition 2.2. Let G be an ample groupoid, A ⊆ G (0) a closed G-invariant set. Then the restriction map CO(G (0) ) → CO(A) extends to a unique homomorphism Γ(G) → Γ(G| A ) with the universal property that for every other homomorphism π : Γ(G) → B such that

CO(A)
9 9 s s s s s commutes, there is a unique extension to a commutative diagram The following converse to Proposition 2.2 can be considered a reformulation of [LV19, Lemma 5.6].
Lemma 2.3. Let G be an ample groupoid and A ⊆ G (0) a closed subset, such that the restriction res A : CO(G (0) ) → CO(A) extends to a map of inverse semigroups Γ(G) → B, for some inverse semigroup B. Then A is G-invariant.

Minimal groupoids and 0-simplifying Boolean inverse semigroups
We introduce the algebraic notion corresponding to minimality of groupoids following [SS20]. The name of the notation 0-simplifying stems from the fact that additive ideals are exactly of the form π −1 (0) for homomorphisms of Boolean inverse semigroups B → C.

CCR and type I groupoids
We refer the reader to [Mur90] for the basic theory of C * -algebras. To each of the objects considered in Sections 2.1 and 2.2 one can associate a C * -algebra. For groupoid C *algebras C * (G), we refer the reader to [Ren80]. The C * -algebras C * (S) associated with inverse semigroups are explained in [Kum84;Pat99]. In particular, it is known that C * (S) ∼ = C * (G(S)) canonically. We refer to [Mur90, Section 5.6] for details on the following two notions from representation theory of C * -algebras.
Definition 2.6. Let A be a C * -algebra. Then A is called CCR if the image of every irreducible *-representation of A equals the compact operators. We call A GCR or type I if the image of every irreducible *-representation of A contains the compact operators.
Inverse semigroups and a groupoids are called CCR or type I, respectively, if their C * -algebras have this property.
In this article, the notions of CCR and type I are accessed solely through the following special case of a result of Clark combined with the characterisation of discrete type I groups by Thoma.

Separation properties of orbit spaces
In this section we consider separation properties of orbit spaces and provide algebraic characterisations of ample groupoids whose orbit space is a (T 1 )-space and a (T 0 )-space, respectively.
Let us start by introducing the simplest example of a groupoid whose orbit space is not (T 1 ). Proof. Assume that the orbit space of G is not (T 1 ). Then by Proposition 2.1 there is some non-closed orbit, say Gx ⊆ G (0) . Let (x n ) n∈N be a convergent sequence from Gx whose limit x ∞ = lim x n does not lie in Gx. Since G is ample, there are bisections (s n ) n∈N in Γ(G) such that s n ∩ d −1 (x n ) ∩ r −1 (x n+1 ) = ∅ for all n ∈ N. Without loss of generality, we may assume that (supp s n ) n are pairwise disjoint subsets of G (0) . Let B ⊆ Γ(G) be the Boolean inverse semigroup generated by all (s n ) n∈N together with the idempotents of Γ(G Assume now that B (T 1 ) is a subquotient of Γ(G). Denote by (s n ) n∈N and f preimages in Γ(G) of the generators of B (T 1 ) . Replacing s n by s n f , we may suppose that supp s n ≤ f holds for all n ∈ N. Further, writing p n = k<n supp s k and replacing s n by s n (f − p n ), we may assume that (supp s n ) n are pairwise orthogonal. Write t n = s n−1 · · · s 0 and q n = t * n (supp s n )t n . Then every finite subfamily of (q n ) n∈N has a non-zero meet, since the same statement holds true for their images in B (T 1 ) . Denoting by U n ⊂ G (0) the compact open subset corresponding to q n , it follows that n∈N U n = ∅. Choose x 0 in there and define x n = t n x 0 ∈ supp s n . Since supp s n ≤ f for all n ∈ N, there is a convergent subsequence (x n k ) k of (x n ). So the orbit Gx 0 has an accumulation point, which proves that the orbit space of G is not (T 1 ).
Remark 3.3. Comparing the statement of Proposition 3.2 with Proposition 3.4, it is natural to ask whether it is possible to find a corner of Γ(G) that has B (T 1 ) as a quotient, rather than finding B (T 1 ) as a subquotient of Γ(G). This is not possible as the following example show. We consider the groupoid G arising from the equivalence relation on βN, the Stone-Cech compactification of N, that relates all elements from N, and nothing else. It is straight forward to check that G is an ample groupoid, since every point of N is isolated in βN. Corners of Γ(G) correspond to restrictions of G to compact open subsets, as explained in Section 2.3. Compact open subsets of βN are exactly of the form U = D for subsets D ⊂ N. Clearly if D is finite, G (T 1 ) cannot arise as a restriction of G| U = G| D . But every infinite subset of N has infinitely many accumulation points in βN, so G (T 1 ) cannot arise as a restriction of G at all.
Proposition 3.4. Let G be a second countable, ample Hausdorff groupoid. Then the following statements are equivalent.
• The orbit space of G is not (T 0 ).
• A corner of Γ(G) has an infinite, monoidal and 0-simplifying quotient.
Proof. Assume first that the orbit space of G is not (T 0 ). By Proposition 2.1, there exists a self-accumulating orbit of G. Denote its closure by A. Then A is a G-invariant subset of G (0) , so that by Proposition 2.2 the restriction to A induces a quotient map Γ(G) → Γ(G| A ). The groupoid G| A has a dense orbit, so that by [Ste19, Lemma 3.4] the set of its units with a dense orbit is comeager. In particular, there is a compact open subset U ⊆ A such that every point of U has a dense G| A -orbit. Note that since U ⊆ A is open, G| U is étale. Further, since U is compact open, Γ(G| U ) is a corner of Γ(G| A ). We note that A is infinite, since it is self-accumulating, so that also U is infinite. Hence Γ(G| U ) is infinite. Further, Γ(G| U ) is monoidal, since U is compact. It is 0-simplifying by Proposition 2.5, since G| U is minimal. If V ⊆ G (0) denotes any compact open subset such that V ∩ A = U, then the quotient map Γ(G) → Γ(G| A ) maps Γ(G| V ) onto Γ(G| U ). So Γ(G| U ) is a quotient of a corner of Γ(G).
If Γ(G) has a corner with an infinite, monoidal and 0-simplifying quotient, then it is a subquotient of Γ(G). So let us assume that there is an infinite, monoidal and 0simplifying subquotient of Γ(G). We will show that the orbit space of G is not (T 0 ). Write Γ(G) ⊃ C ։ B for the given subquotient. Choosing an idempotent preimage p ∈ E(C) of the unit of B, we obtain a surjection pCp ։ B. Let V ⊂ G (0) be the compact open subset corresponding to p. Replacing C by pCp, we thus find a unital inclusion Γ(G| V ) ⊃ C and a quotient map π : C ։ B. Let H be the ample groupoid associated with C by noncommutative Stone duality, that is Γ(H) ∼ = C. Write X = H (0) . Considering the restriction of π to idempotents, we find a closed subset A ⊆ X such that the following diagram commutes.
We will show that Hx ∩ A is self-accumulating. By noncommutative Stone duality, there is an ample groupoid K such that Γ(K) ∼ = B. From the fact that B is infinite, monoidal and 0-simplifying, it follows that K has is infinite, has a compact unit space and is minimal. In particular, the orbits of K are self-accumulating. So for every compact open neighbourhood U ⊆ A of x, there is some bisection t ∈ Γ(K) such that x ∈ supp t and x / ∈ im t ≤ U. Let s ∈ π −1 (t) denote a preimage of t. Then supp s ∩ A = π(supp s) = supp π(s) = supp t and similarly im s ∩ A = im t. It follows that Hx ∩ A and thus also the H-orbit of x is self-accumulating. Thus the orbit space of H is not (T 0 ). Consider now the surjective map ϕ : V → X which is dual to the inclusion CO(X) ⊂ CO(V ). Given x, y ∈ X in the same H-orbit, there is s ∈ C such that sx = y. Let u ∈ V be some preimage of x under ϕ. Considering s as a bisection of G| V , we define v = su, which lies in the same G| V -orbit as u and satisfies ϕ(v) = y. This shows that every H-orbit is contained in the image of a G| V -orbit. In particular, there is some orbit of G| V that is not finite and hence not locally closed. So Proposition 2.1 says that the orbit space of G is not (T 0 ).

Group quotients and isotropy groups
In this section, we relate the isotropy groups of an ample groupoid with certain subquotients of the Boolean inverse semigroup of its compact open bisections. This result will allow us to address the condition on isotropy groups from [Cla07].
Proposition 4.1. Let G be a second countable, ample Hausdorff groupoid whose orbit space is (T 0 ). Let x ∈ G (0) be a unit, write G = G| x for the isotropy group at x and denote by G 0 the associated group with zero. Then G 0 is a quotient of a corner of Γ(G). Vice verse, if G is any ample Hausdorff groupoid and G is a group such that G 0 is a quotient of a corner of G, then G is a quotient of a point stabiliser of G.
Proof. Since the orbit space of G is assumed to be (T 0 ) and G is second countable, its orbits are locally closed by Proposition 2.1. Let U ⊆ G (0) be a compact open neighbourhood of x, such that Gx ∩ U is closed in U and hence compact. Since G is étale, we know that Gx is countable, so that Gx ∩ U is actually finite. We may thus shrink U so that Gx ∩ U = {x} holds. Denote by p = U ∈ Γ(G) the idempotent bisection associated with U. Then the corner of Γ(G) can be identified as pΓ(G)p = Γ(G| U ). Since x is fixed by G| U , the restriction from U to {x} induces a quotient of Boolean inverse semigroups Γ(G| U ) → Γ(G| x ) = (G| x ) 0 by Proposition 2.2.
Let us know assume that G is any ample Hausdorff groupoid, let G be a group and p ∈ Γ(G) an idempotent, for which there is a quotient map π : pΓ(G)p ։ G 0 . Denote by U ⊆ G (0) the compact open subset corresponding to p. Then there is a natural isomorphism pΓ(G)p ∼ = Γ(G| U ). Since the algebra of idempotents of G 0 is trivial, π| E(Γ(G| U )) is a character. By Stone duality, there is x ∈ U such that π| E(Γ(G| U )) = ev x . In particular, {x} ⊆ U is a G| U -invariant subset by Lemma 2.3. So by the universal property of the restriction map described in Proposition 2.2, the homomorphism π factors through Example 4.2. It might be tempting to admit arbitrary subquotients of Γ(G) in the statement of Proposition 4.1, however this does not even suffice under the condition that the orbit space of G is (T 1 ). Indeed, the topological full group of G is always a subgroup of Γ(G), and it can be large even if G is effective. For example the topological full group associated with B (T 1 ) is Sym(N).
We next formulate an appropriate version of Proposition 4.1 for inverse semigroups. Let us start with a short lemma relating quotients of an inverse semigroup and its booleanization.
Lemma 4.3. Let S be an inverse semigroup and B(S) ։ G 0 a quotient of its booleanization. Then S 0 → G 0 is surjective.
Proof. Denote the quotient map B(S) ։ G 0 by π and let g ∈ G. Using the description of B(S) presented in Section 2.1, there is some preimage i t i e i ∈ B(S) of g. Since G 0 has only two idempotents, π| E(B(S)) is a character. So there is a unique i 0 satisfying π(e i 0 ) = 1. This implies π( i t i e i ) = π(t i 0 ), showing that g ∈ π(S 0 ). Proposition 4.4. Let S be a countable inverse semigroup such that the orbit space of G = G(S) is (T 0 ). Let x ∈ G (0) and write G = G| x . Then the group with zero G 0 is a quotient of a corner of S 0 .
Proof. Fix x ∈ G (0) . Since G (0) ∼ = E(S), there is q ∈ E(S) such that x(q) = 1. Denote by U ⊆ G (0) the compact open subset corresponding to q. As in the proof of Proposition 4.1, the fact that orbits of G are locally closed implies that Gx ∩ U is finite. Fix an enumeration x = x 0 , x 1 , . . . , x n of Gx ∩ U. For i ∈ {1, . . . , n} there is q i ∈ E(S) such that x 0 (q 0 ) = 1 and x i (q i ) = 0. Put p = q · q 1 · · · q n and let V ⊆ G (0) be the compact open subset corresponding to p. Then the identification of corners of Boolean inverse semigroups says that G(pSp) = G(B(pSp)) ∼ = G(pB(S)p) ∼ = G| V .

Proof of the main results
We now obtain the proof of our main theorems. Thanks to the preparation made in the previous sections, all proofs are rather similar and we spell out details only for Theorem A.