Analyzing the Weyl construction for dynamical Cartan subalgebras

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$.

One of the earliest theorems about C * -algebras, the Gelfand-Naimark Theorem, establishes that any commutative C * -algebra is of the form C 0 (X) for a locally compact Hausdorff space X. In addition to inspiring the "noncommutative topology" approach to C * -algebras, the Gelfand-Naimark Theorem has also led researchers to search for Abelian subalgebras inside noncommutative C * -algebras, with the goal of using topological tools to analyze the Abelian subalgebra, and from there to obtain a better understanding of its noncommutative host. This program has been particularly successful when the subalgebra is Cartan (see Definition 2.1 below); it has enabled progress on Elliott's classification program for C * -algebras [BL17,Li19] as Date: October 9, 2020. well as the theory of continuous orbit equivalence of topological dynamical systems [MM14,BCW17]. Even beyond the setting of Cartan subalgebras, many authors (cf. [BCW17, BFPR19, IKR + 20]) have successfully extended structural information from more general Abelian subalgebras to the containing C * -algebras.
In this paper, we focus our attention on certain Cartan subalgebras which appear in a rather unexpected context. Renault proved in [Ren08], building on earlier work of Kumjian [Kum86], that if a C * -algebra A admits a Cartan subalgebra, then A is isomorphic to the reduced C * -algebra C * r (G, Σ) of a twist Σ over a groupoid G, and the Cartan subalgebra is realized as C 0 (G (0) ). The groupoids G appearing in Renault's analysis must satisfy a number of structural contraints; for example, they are always topologically principal. If G is not topologically principal, then C 0 (G (0) ) is not a Cartan subalgebra inside C * r (G, Σ) for any twist Σ over G. Nevertheless, there are many such groupoids whose twisted C * -algebras contain Cartan subalgebras. Examples include the rotation algebras A θ ∼ = C * r (Z 2 , c θ ) and the C * -algebras of directed graphs which do not satisfy Condition (L).
Together with Reznikoff and Wright, in [DGN + 20, Theorem 3.1] we identified a large family of twisted groupoid C * -algebras, associated to non-principal groupoids G, which contain Cartan subalgebras. Moreover, these Cartan subalgebras are evident at the level of the groupoid G: they arise from a subgroupoid S of G. The existence of a Cartan subalgebra in C * r (G, c) implies, by [Ren08], the existence of a topologically principal groupoid G, the so-called Weyl groupoid, and a twist Σ over G such that C * r (G, c) ∼ = C * r (G, Σ). If G is a discrete group, and S ≤ G satisfies the hypotheses of [DGN + 20, Theorem 3.1] so that C * r (S, c) is a Cartan subalgebra of C * r (G, c), then [DGN + 20, Theorem 5.2] establishes that G = (G/S) ⋉ S as long as the action of G/S on S is topologically free. Moreover, [DGN + 20, Theorem 5.8] shows that in this case, the twist Σ arises from a 2-cocycle on G, which we described explicitly in [DGN + 20, Lemma 5.6].
The preprint [IKR + 20] came to our attention as we were finalizing [DGN + 20], and we were struck by the structural parallels between the two papers' main results. In [IKR + 20,Theorem 3.4], the authors show that if a subgroupoid A of a groupoid G consists of a closed normal bundle of Abelian groups, then C * r (G) ∼ = C * r ( A ⋊ (G/A), Σ). However, they only establish that C * r (A) is Cartan in C * r (G) if G isétale and G/A is topologically principal [IKR + 20, Theorem 5.6], and they do not analyze the structure of twisted groupoid C * -algebras C * r (G, c). As non-trivial discrete groups are never topologically principal, this excludes the setting of [DGN + 20, Theorem 5.8]. Moreover, the formula given in [IKR + 20] for the twist Σ is not explicit. In particular, it is unclear when, or whether, the twist Σ can be realized via a 2-cocycle on the groupoid A ⋊ (G/A).
In this paper, we bridge the gap between [DGN + 20] and [IKR + 20]. Our first main result, Theorem 4.7, establishes that when a subgroupoid S of anétale groupoid G satisfies the hypotheses of [DGN + 20, Theorem 3.1], so that C * r (S, c) is Cartan in C * r (G, c), then the Weyl groupoid associated to the Cartan pair (C * r (G, c), C * r (S, c)) is an action groupoid (G/S) ⋉ B, where B denotes the spectrum of the commutative algebra C * r (S, c). When the 2-cocycle c is trivial, B agrees with the space S of [IKR + 20], and (translating our left action of G/S into a right action) we see that our groupoid (G/S) ⋉ B agrees with the groupoid S ⋊ (G/S) of [IKR + 20]; see Remark 4.8. Theorem 4.7 is a substantial improvement over [DGN + 20, Theorem 5.2]. Not only do we extend [DGN + 20, Theorem 5.2] from groups to groupoids, we also show that the hypothesis of topological freeness in the latter theorem is always satisfied.
Our second main result, Theorem 6.1, shows in particular that if the quotient map q : G → G/S admits a continuous section s, then the Weyl twist Σ of the Cartan pair (C * r (G, c), C * r (S, c)) is given by a continuous 2-cocycle C s on (G/S) ⋉ B. The formula for C s is given in Proposition 5.1. Our precise description of the twist Σ represents both an extension of [DGN + 20, Theorem 5.8] to a broader setting and an improvement on [IKR + 20, Theorem 3.4] in theétale case.
A detailed analysis of the Weyl groupoid underlies our results in this paper, and we expect that some of the technical results we have obtained, such as Proposition 2.3 and Corollary 3.11, will also be of general interest. Proposition 2.3 gives a description of the equivalence relation underlying the Weyl groupoid which, to our knowledge, has not appeared before in the literature. Corollary 3.11 describes the spectrum of the twisted groupoid C * -algebra C * r (S, c) if S is a bundle of Abelian groups and c is symmetric on S. This description may be known to experts -indeed, it is similar to results such as [MRW96, Corollary 3.4], [Goe08, Proposition 5], and [CN20, Remark 5.2] -but we were unable to locate a reference in the literature for the precise result we needed. This paper is organized as follows. We recall the relevant definitions of groupoids and Cartan subalgebras in Section 2, and we establish Proposition 2.3. The first step in providing an explicit description of the Weyl groupoid associated to a Cartan pair (A, B) is identifying the topological space B; Section 3 is devoted to describing B in the case when B = C * r (S, c) arises from a bundle of discrete Abelian groups. Section 4 contains our first main result. We show in Theorem 4.7 that if (A, B) = (C * r (G, c), C * r (S, c)) is one of the Cartan pairs identified by [DGN + 20, Theorem 3.1], then its Weyl groupoid is an action groupoid, arising from an action of the quotient groupoid G/S on the spectrum B ∼ = B. Given a section s : G/S → G of the quotient map, we describe in Section 5 a 2-cocycle C s on the action groupoid (G/S) ⋉ B. When s is continuous, we prove in Theorem 6.1 that C s gives the Weyl twist over

Preliminaries on Cartan subalgebras and the Weyl construction
Intuitively, a groupoid G is a generalization of a group in which multiplication is only partially defined. More precisely, a groupoid is a set G, together with a subset G (2) ⊆ G × G; a multiplication map (γ, η) → γη from G (2) to G; and an inversion map γ → γ −1 from G to G, which behave like multiplication and inversion do in groups wherever they are defined. The unit space of G is G (0) = {γγ −1 : γ ∈ G}. We then have range and source maps r, s : G → G (0) given by which satisfy r(γ)γ = γ = γs(γ) for all γ ∈ G. Given u ∈ G (0) , we write G u := {γ ∈ G : s(γ) = u} and G u := {γ ∈ G : r(γ) = u}. We can also describe G (2) using the range and source maps: In this paper we will assume that G is equipped with a locally compact Hausdorff topology with respect to which the multiplication and inversion maps are continuous. The groupoids considered in this paper will also beétale -that is, r and s will be local homeomorphisms.
The link between groupoids and Cartan subalgebras was established in the seminal papers [Kum86,Ren08].
(1) B is a maximal Abelian subalgebra of A.
(3) B is regular; that is, the normalizer of B, N (B) := {n ∈ A such that nbn * , n * bn ∈ B for all b ∈ B}, generates A as a C * -algebra. (4) B contains an approximate identity for A.
For this section, we fix a Cartan subalgebra B of some separable C * -algebra A. Let us first recall how (A, B) gives rise to a topologically principal groupoid and twist (cf. [Kum86,1.6] or [Ren08, Proposition 4.7]), and then gather a few tools to study them. Let B be the spectrum of B, viewed as the space of one-dimensional representations of B (a subspace of B * , the space of linear functionals on B), and let Ω : C 0 ( B) → B be the Gelfand representation.
The Weyl groupoid G (A,B) is the quotient of We will denote the equivalence class of (α n (x), n, x) by [α n (x), n, x]. The groupoid structure on G (A,B) is defined by To topologize G (A,B) , we define a basic open set to be of the form In particular, for b = nn * , the fact that each functional x ∈ B is multiplicative implies that Since x ∈ dom(n), we can divide by x(n * n) and get 0 = x(n * n) = α n (x)(nn * ) as claimed.
The Weyl twist Σ (A,B) is another groupoid associated to the Cartan pair (A, B). Like the Weyl groupoid, the Weyl twist is also a quotient of D, but by the equivalence relation (α n (x), n, x) ≈ (α m (y), m, y) ⇐⇒ We write α n (x), n, x for the class of (α n (x), n, x) in Σ (A,B) . We point out that equivalence with respect to ≈ implies equivalence with respect to ∼.
As its name suggests, the Weyl twist is a T-groupoid, or twist, over G (A,B) . As such, one can construct the twisted groupoid C * -algebra C * r G (A,B) , Σ (A,B) (cf. [Ren08, Section 4]). The Weyl twist and groupoid are constructed exactly so that , see [Ren08, Theorem 5.9]. We recall the construction of a twisted groupoid C * -algebra in the case when the twist arises from a 2-cocycle, as this is the level of generality we will need in this paper. Recall that a (T-valued ) 2-cocycle on a groupoid G is a function c : G (2) → T which satisfies the cocycle condition c(x, yz) c(y, z) = c(x, y) c(xy, z) for all (x, y), (y, z) ∈ G (2) .
Given a second countable, locally compact Hausdorff,étale groupoid G and a continuous 2-cocycle c : G (2) → T, we denote by C c (G, c) the collection of continuous C-valued functions on G which we view as a * -algebra via the twisted convolution multiplication f g (γ) := ηρ=γ f (η)g(ρ)c(η, ρ) and the involution f * (γ) := f (γ −1 )c(γ, γ −1 ). For each u ∈ G (0) , we represent C c (G, c) on ℓ 2 (G u ) by left multiplication: Let f u denote the operator norm of π u (f ). The reduced twisted groupoid C *algebra C * r (G, c) is then the completion of C c (G, c) in the norm · r := sup u∈G (0) · u . While the definition of the Weyl groupoid and Weyl twist seem quite different, the following very helpful proposition describes the groupoid in terms more similar to the twist. The result may be known to experts, but we were unable to locate it in the literature. For the proof, we require two lemmata.
Lemma 2.5 (Urysohn-type Lemma). Let f ∈ N (B) and suppose that k ∈ C 0 ( B) has supp ′ (k) ⊆ dom(f * ). Then the partial homeomorphism associated to Ω(k)f has domain α −1 f (supp ′ (k)) = α f * (supp ′ (k)), where it coincides with α f . Proof. First note that f 2 := Ω(k)f is still a normalizer of B because f is and because Ω(k) ∈ B; thus, it makes sense to speak of the corresponding partial homeomorphism α f2 and its domain, dom(f 2 ).
By our construction of X, we have y(b i ) = 0 and also X ⊆ dom(n i ), so that y(n * i n i ) > 0. We have shown that, for any g ∈ C 0 ( B), g (α n1 (y)) is a positive multiple of g (α n2 (y)). Since C 0 ( B) separates points (as B is Hausdorff), this implies α n1 (y) = α n2 (y) for all y ∈ X. As X is an open neighborhood of x, we arrive at the claimed equality in the Weyl groupoid.

The spectrum of a twisted bundle of groups
Assume that S is a second countable, locally compact Hausdorff,étale groupoid and that c : S (2) → T is a 2-cocycle on S. We will always assume that 2-cocycles are normalized, i.e. c(r(a), a) = 1 = c(a, s(a)) for each a ∈ S. In order to construct the twisted groupoid C * -algebra C * r (S, c), we will need c to be continuous, so we will frequently impose this assumption.
In this section, we will be interested in bundles of groups, so on top of our topological assumptions above, assume that the range and source maps of S are equal, called p : S → S (0) . We write S u := p −1 ({u}) for u a unit. Moreover, assume that the multiplication map S (2) → S is commutative and that the continuous 2cocycle c is symmetric on S, i.e. c(a, a ′ ) = c(a ′ , a) for all a, a ′ ∈ S u , so that its reduced twisted C * -algebra B := C * r (S, c) is commutative by [DGN + 20, Lemma 3.5].
Remark 3.1. Observe that S is amenable. Indeed, since S is a bundle of Abelian groups, C * r (S) is commutative; in particular, it is nuclear. Since S isétale, amenability therefore follows from [Sim19, Thm. 4.1.5]. In particular, [BaH14, Corollary 4.3] implies that C * (S, c) ∼ = C * r (S, c). Definition 3.2. Given u ∈ S (0) and a continuous 2-cocycle c on S, let B u denote the set of 1-dimensional c-projective representations of the Abelian group S u . That is, B u consists of the maps χ : S u → T such that (Observe that such maps χ necessarily satisfy χ(u) = 1, because c(a, u) = 1 = c(u, a) for all a ∈ S u .) Write B = u∈S (0) B u and ρ : B → S (0) for the projection map.
Recall that the spectrum C of a commutative C * -algebra C is the set of nondegenerate one-dimensional representations of C, equipped with the weak- * topology.
Then φ χ is a * -algebra homomorphism and extends to an element of B.
At times, we will write φ(χ) instead of φ χ , to ease notation.
Remark 3.4. It is unclear to the authors whether the formula for φ χ in Lemma 3.3 extends to elements of B when thought of as C 0 -functions on S.
Proof. First note that since S isétale and second countable, S u is countable, so φ χ (f ) is a finite sum for any χ ∈ B and any f ∈ C c (S, c).
It is evident that φ χ is linear. To see that φ χ is multiplicative, note that if u := ρ(χ), then Using the defining formula (8) for χ, To see that φ χ is * -preserving, we use Equation (8) to compute To see that φ χ : C c (S, c) → C extends to an element of B, i.e. a non-zero normdecreasing * -homomorphism φ χ : In our case, where S = Iso(S), we have We have already shown that each φ χ is a *-representation of C c (S, c), and by the triangle inequality, it is I-norm decreasing, i.e. one of the representations invoked in Equation (10). In particular, φ χ (f ) ≤ f for any f ∈ C c (S, c) and any χ ∈ B, so the representation φ χ extends to a nonzero * -homomorphism Therefore, to show that φ is a bijection, it suffices to show that all one-dimensional representations of C * r (S u , c) are of the form φ χ for some χ ∈ B. To that end, we observe that B is a C 0 (S (0) )-algebra (cf. [CN20, Remark 5.1]). Then [Wil07, Proposition C.5] implies that B = u∈S (0) B(u). When c is trivial, [CN20, Remark 5.2] and Remark 3.1 imply that B(u) ∼ = C * r (S u , c). In fact, a careful examination of the proof of [CN20, Remark 5.2] reveals that, even if c is not trivial, the formulae used there will also give an isomorphism between the fiber algebra B(u) and the twisted group C * -algebra C * r (S u , c). It is a classical fact that representations of the twisted group C * -algebra C * (S u , c) ∼ = C * r (S u , c) ∼ = B(u) are in bijection with unitary projective representations of S u . In particular, the one-dimensional representations of B(u) are of the form for some χ ∈ B u , see the formula in [BS70, Theorem 3.3(2)]. Comparing the above with the definition of φ, we see that indeed, the map χ → φ χ is surjective, and hence a bijection.
Proposition 3.6. If we equip B with the topology induced by φ from B, then a net (χ i ) i converges to χ in B if and only if the following two conditions hold: Remark 3.7. Note that, with respect to this topology on B, ρ : B → S (0) is clearly continuous. Note further that, when c is trivial, this result is well-known; the description of the topology on B should be compared to [MRW96,Proposition 3.3].
The proof of Proposition 3.6 proceeds through a series of lemmata.
is an upper bound for j 1 and j 2 in J. Firstly, this implies that (u j ) j∈J is a subnet of (u i ) i∈I (the inclusion J ֒→ I is monotone and final), so that Condition (1) of Proposition 3.6 implies lim j∈J u j = lim i∈I u i = ρ(χ). Secondly, we conclude that (a j ) j∈J is a net in supp ′ (f ). Since supp(f ) is compact, there exists a subnet (a κ ) κ∈K of (a j ) j∈J which converges to some a ∈ supp(f ). By continuity of f , we have i.e. a ∈ supp ′ (f ). But p(a) = lim κ p(a κ ) = lim κ u κ = ρ(χ), i.e. a ∈ S ρ(χ) also, which contradicts our hypothesis that supp ′ (f ) ∩ S ρ(χ) = ∅.
Proof. We must show that, for all f ∈ C c (S, c) and for all ǫ > 0, there We will begin by proving the claim for f ∈ C c (S, c) such that supp(f ) is a bisection. Let u := ρ(χ) and u i : If we now use the fact that f ∈ C c (S, c) is bounded in · ∞ and our hypothesis that (χ i ) i and χ satisfy Condition (2) of Proposition 3.6, an easy ǫ/2-argument establishes that |φ χi (f ) − φ χ (f )| < ǫ for i ≥ i 0 for some i 0 ∈ I.
For more general functions, recall that since S is a second countable, locally compact Hausdorff,étale groupoid, we have Lemma 3.1.3]. An ǫ/k-argument now shows that, for any g ∈ C c (S, c), there exists i 1 ∈ I so that i ≥ i 1 implies |φ χi (g) − φ χ (g)| < ǫ.
Proof. Recall that our assumption φ χi → φ χ means that, for every f ∈ C c (S, c) and We start by proving (1) As S (0) ⊇ V is a bisection containing supp ′ (f ), and ξ(v) = 1 for any ξ ∈ B and v ∈ S (0) , the above inequality becomes This concludes the proof of (1).
We proceed with proving (2). Suppose (a i ) i ⊆ S such that p(a i ) = u i , p(a) = u, and a i → a. Fix ǫ > 0. We must show there exists M ∈ Λ such that if i ≥ M , then |χ i (a i ) − χ(a)| < ǫ.
Again by [Fol99,4.32 (Urysohn's Lemma)], there exists f ∈ C c (S, c) which is equal to 1 on U and 0 outside of W . So for all i in Λ which are larger than both N and N f,ǫ , we know a i ∈ U ⊆ supp(f ) and Note that W is a bisection, and a i , a are elements of U ⊆ W with p(a i ) = u i and p(a) = u. All of these facts combined yield that a i is the unique element in S ui ∩ U and a is the unique element in S u ∩ U . Since f is equal to 1 on U , the inequality becomes |χ i (a i ) − χ(a)| < ǫ. This completes the proof of the lemma, and of Proposition 3.6.
Corollary 3.11. The map B → B, χ → φ χ , is a homeomorphism when B is equipped with the topology described in Proposition 3.6. In particular, B is locally compact Hausdorff and B is isomorphic to the C * -algebra C 0 (B).

Computing the Weyl groupoid
Our standing assumptions for the remainder of this paper are the following: Let us explain why our last assumption on S is reasonable. It was shown in [DGN + 20, Theorem 3.1] that, in order to get (e), a sufficient assumption on S is that (1) S is maximal among the Abelian subgroupoids of Iso(G) on which c is symmetric, and additionally (2) S is immediately centralizing (see [DGN + 20, Definition 2]). A careful examination of the proof of [DGN + 20, Proposition 3.9] reveals that, instead of (2), we may assume that for each η ∈ Iso(G) with u = r(η) = s(η), the set {aνa −1 | a ∈ S u } is either the singleton {ν} or infinite.
In the sections about to come, we will use the techniques we have developed so far to compute the Weyl groupoid G (A,B) and the Weyl twist Σ (A,B) of the Cartan pair (A, B). In particular, we will see in Theorems 4.7 and 6.1 that there is a strong connection to a certain groupoid action of G/S on B. As such, it seems prudent to briefly state a few facts about the quotient groupoid G/S. Remark 4.1. We let q : G → G/S =: Q, γ → q(γ) =:γ, denote the quotient map. Since S is a wide subgroupoid of G (i.e. S ⊆ Iso(G) is closed with S (0) = G (0) ), openness of S andétaleness of G imply that q is an open map. Since G is Hausdorff and S is closed in G, this implies that Q is Hausdorff also. Furthermore, it follows from [Wil19, Corollary 2.13] (taking G = S and X = G) that Q is locally compact and second countable because G is.
Lastly we point out that, if one is interested in groupoids S ⊆ G withétale quotient Q (as in [IKR + 20, Theorem 5.6]), then one must ask for S to be open in G, as we have done.
We will now construct a continuous left actionα of the locally compact Hausdorff groupoid Q = G/S on the spectrum B of B = C * r (S, c), with the moment map ρ : B → Q (0) given by ρ| Bu = u. In the following, we will write Q * ρ B := {(γ, χ) ∈ Q × B | s Q (γ) = ρ(χ)}.
In other words,α is a continuous left action of Q on B with moment map ρ.
Before embarking on the proof, we point out that the formula forα is not surprising. Indeed, if χ were defined on all of G and satisfied Equation (8) (and, by extension, Equation (9)), then we would havẽ Proof. One readily verifies (1)-(3) using the cocycle identity (Equation (2)), the fact that c is symmetric on the Abelian subgroupoid S, that c is normalized, and that c(γ −1 , γ) = c(γ, γ −1 ) for any γ ∈ G by [DGN + 20, Lemma 2.1].
For (4), suppose that (γ i , χ i ) → (γ, χ) in Q * B. We need to show (see [MRW96,Proposition 3.3]) that, if a i → a in S and s(a i ) = r(γ i ) for all i, thenαγ i (χ i )(a i ) → αγ(χ)(a). Since the cocycle c and both multiplication and inversion on G are continuous, we have The actionα of Q on B allows us to endow the space Q * ρ B with the structure of a topological groupoid. This so-called left action groupoid is denoted Q ⋉ B, and we will show in Theorem 4.7 that it is isomorphic to the Weyl groupoid G (A,B) .
Our next goal will be to describe the relationship between the partial homeomorphisms α n used to construct the Weyl groupoid G (A,B) and the actionα (see Proposition 4.5 below). We begin with a few preliminary results. ( with s(γ 1 ) = s(γ 2 ) = u, then either of the above assumptions impliesγ 1 =γ 2 . Proof. We start by proving (2). By assumption, φ χ ∈ dom(f 1 ) ∩ dom(f 2 ) and As the conditional expectation Φ fixes B and is B-linear, we get the equality in the following: It follows that φ χ Φ(f * 2 f 1 ) > 0, as claimed. For (1), we use Proposition 2.3 to obtain b i ∈ B such that φ χ (b i ) = 0 and f 1 b 1 = f 2 b 2 . The above proof now works mutatis mutandis, replacing each instance of '> 0' by ' = 0'.
The fact that α f (φ χ ) = φαγ (χ) whenever {γ} = supp ′ (f ) ∩ G ρ(χ) for f ∈ N and χ ∈ B, shows that there is an intimate connection between the partial action α of N (B) on B and the actionα of Q on B. In order to describe this connection, we first need to better understand equality in the Weyl groupoid G (A,B) .
Proposition 4.6. Suppose f i ∈ N and let X i := supp ′ (f i ) ⊆ G and χ ∈ B. Then the following are equivalent: In the above, we wrote X i u for the singleton-set X i ∩ G u , and q : G → Q for the quotient map.
Theorem 4.7. There is an isomorphism ϕ of topological groupoids Q⋉B → G (A,B) given by where f ∈ C c (G, c) is any function supported on a bisection such thatγ ∈ q(supp ′ (f )).
We point out here that this result is a significant strengthening of [DGN + 20, Theorem 5.2]. Not only is Theorem 4.7 true forétale groupoids (not just discrete groups), but we also do not need to assume thatα is topologically free. Instead, as our theorem suggests, this simply follows from G (A,B) being topologically principal. Moreover, Theorem 4.7 applies in the setting of [IKR + 20, Section 3] if one assumes the groupoids involved to beétale, as discussed in the following remark. is precisely the left actionαγ−1(φ χ ) ofγ −1 ∈ G/S on φ χ in this case. In other words, Theorem 4.7 establishes that the groupoid S ⋊ Q of [IKR + 20, Theorem 3.4] is indeed the Weyl groupoid, if G isétale and C * r (S) is Cartan in C * r (G). We will use the rest of this section to prove Theorem 4.7 through a series of lemmata. Proof. Let (γ, χ) ∈ Q ⋉ B and f ∈ N satisfyγ ∈ q(supp ′ (f )). This assumption implies ρ(χ) = s(γ) ∈ s(supp ′ (f )), which guarantees that φ χ ∈ dom(f ) by Proposition 4.5(1), so that [φαγ (χ) , f, φ χ ] is indeed an element of G (A,B) . Moreover, this element is independent of the choice of f by Proposition 4.6, (2) =⇒ (3). In other words, ϕ is well-defined.
Let U be any open neighborhood around τ contained in supp ′ (f ). Then, since G isétale, s(U ) is an open neighborhood around s(τ ) = ρ(χ). As χ i → χ and ρ is continuous, there exists i 0 so that for all i ≥ i 0 , we have s(τ i ) = ρ(χ i ) ∈ s(U ). Since τ i ∈ supp ′ (f ) and U = s −1 (s(U )) ∩ supp ′ (f ), this means τ i ∈ U for i ≥ i 0 . As U was an arbitrary neighborhood around τ , we conclude that τ i converges to τ .
All in all, we have shown that (γ i , χ i ) converges to (τ , χ), the pre-image of Γ under ϕ. This completes the proof of the lemma, and also of Theorem 4.7.

A cocycle on the Weyl groupoid
The standing assumptions for this section can be found on page 13. In this section, we show that every (continuous) section s of the quotient map q induces a (continuous) T-valued 2-cocycle C s on the Weyl groupoid. Though the formula for C s is complicated, it is reminiscent of [Ren08, Lemma 5.3] and will be exactly what we require to prove in Theorem 6.1 that (Q ⋉ B) × C s T and the Weyl twist Σ (A,B) are isomorphic (even as topological groupoids, when s is continuous).
Proposition 5.1. Let s : G/S = Q → G be a section for q : G → Q and for eacḣ γ ∈ Q, choose one fγ ∈ N such that fγ(s(γ)) > 0. The following defines a T-valued 2-cocycle on Q ⋉ B: where Φ : A → B is the conditional expectation. If s is continuous, then C s is continuous.
We will deal with the algebraic and topological aspects of the proof separately.
Lemma 5.2. The map C s from Proposition 5.1 is T-valued and satisfies the cocycle condition of Equation (2).
We need to check that In the following computation, we write ∝ to denote that the left-hand side is a positive multiple of the right-hand side. If we letC denote the numerator of C, then our claim becomes where we threw in the positive number φ χ (f * γ3 fγ 3 ) to eventually make up for the following annoyance: out of the fourC-factors,C(h 1 , h 2 ) is not given by evaluating at φ χ but atαγ 3 (χ) = α fγ 3 (φ χ ).
We computeC , where we used in ( †) the defining property of α f3 , and in ( ‡) that nΦ(m)n * = Φ(nΦ(m)n * ) = Φ(nmn * ) for functions n, m ∈ C c (G, c) with n supported on a bisection: the first equality holds since the left-hand side is an element of B, and the second equality can be checked by hand, using that S ⊆ Iso(G) and that S is normal.
We can now rephrase Equation (15) to ) .
Since F i ∈ C c (G, c) for each i, by definition of φ χ and Φ we have Proof. Let C := C s . Suppose we are given a convergent net in (Q ⋉ B) (2) , say The assumption that lim i (x i , y i ) = (x, y) means that lim iγ i =γ and lim iτ i =τ in Q, and lim Recall that the latter implies that u i := ρ(χ i ) converges to ρ(χ) =: u.
Since q(γ i τ i ) =γ iτi i →γτ = q(γτ ) and s is continuous, we also have In order to show that lim i C(x i , y i ) = C(x, y), consider F i := f * q(γiτi) fγ i fτ i and recall that C(x i , y i ) = φχ i (Φ(Fi)) |φχ i (Φ(Fi))| . The fact that F i is a product of elements of N and hence is supported in a bisection means that S ui ∩ supp ′ (F i ) consists of the single point . We compute (cf. the proof of Proposition 5.1) that In particular, since we assumed that fκ(s(κ)) > 0 for anyκ ∈ Q, we have . As γ i → γ, τ i → τ , and γ i a i τ i → γaτ (all by continuity of s), we have that a i → a. Since χ i → χ, it follows that Note that the right-hand side is, by the same argument as for C(x i , y i ), exactly C(x, y). This concludes our proof.

Computing the Weyl twist
Our standing assumptions can be found on page 13. In the main result of this section (Theorem 6.1), we prove that every section s of the quotient map q : G → S induces a T-equivariant isomorphism of the (discrete) groupoids (Q ⋉ B) × C s T and Σ (A,B) . Moreover, if s is continuous, then the isomorphism is a homeomorphism, meaning we have explicitly identified not only the Weyl groupoid but also the Weyl twist of the Cartan pair (A, B), see Corollary 6.2.
Moreover, if s is a continuous section, then ψ s is a homeomorphism.
Corollary 6.2. If s is continuous, then the Cartan pair (C * r (G, c), C * r (S, c)) is isomorphic to the pair C * r ((Q ⋉ B) × C s T), C 0 (B) . Proof. Since the Weyl groupoid is isomorphic, as a topological groupoid, to Q ⋉ B by Theorem 4.7, and the Weyl twist is also topologically isomorphic to (Q⋉B)× C s T by Theorem 6.1, [Ren08, Theorem 5.9] implies that C * r (G, c) is isomorphic to C * r (Q⋉ B, (Q ⋉ B) × C s T), and that isomorphism carries C * r (S, c) onto C 0 ((Q ⋉ B) (0) ) ∼ = C 0 (B).
The remainder of this section is devoted to proving Theorem 6.1. In order to do so, we will need the following analogue of [DGN + 20, Lemma 5.4] that states that every element of the Weyl twist can be represented by a function supported in a bisection and scaled by an explicitly computed factor. Proposition 6.3. Let α n (φ χ ), n, φ χ be an arbitrary element of the Weyl twist Σ (A,B) , and letγ ∈ Q be such that ϕ(γ, χ) = [α n (φ χ ), n, φ χ ]. If f ∈ N satisfieṡ γ ∈ q(supp ′ (f )), then Note that by the surjectivity of ϕ (see Theorem 4.7), such an elementγ always exists. Moreover, Proposition 4.5(2) shows that α f (φ χ ) = φαγ (χ) for any f as specified in the proposition.
Proof. Write x := φ χ ∈ dom(n). We have to find two elements b, b ′ ∈ B such that x(b), x(b ′ ) > 0 and nb = (λf )b ′ . By assumption on f and definition of ϕ, we have Proposition 2.3 therefore tells us that there exist b 1 , b 2 ∈ B such that f b 1 = nb 2 and x(b 1 ), x(b 2 ) = 0. In fact, the construction of b 1 , b 2 in the proof of that proposition gives x(b 2 ) > 0. Thus, if we set To see that λ can be equivalently written as in the statement of the proposition, recall from Equation (6) that where k ∈ C 0 ( B) is supported on dom(f ) ∩ dom(n) and satisfies k(x) = 1. As the conditional expectation Φ : A → B is B-linear, it follows that This yields the asserted expression for λ.
As we will frequently need to explicitly compute the constant λ appearing in Proposition 6.3, the following corollary will be helpful.
The proof of Theorem 6.1 now proceeds through a series of lemmata. To simplify notation, we will write ψ for ψ s and C for C s in the proofs, tacitly using the chosen s.
Lemma 6.5. The map ψ s is surjective and does not depend on the choice of fγ.
Lemma 6.6. The map ψ s is injective.
Lemma 6.7. The map ψ s is a T-equivariant groupoid homomorphism.
Lemma 6.8. If s is a continuous section, the map ψ s is continuous.
Proof. Let ψ := ψ s and C := C s . Assume Since we also have f i (γ i ) > 0, Corollary 6.4 implies that for i ≥ i 1 , To show that ψ(x i ) is an element of U(U, V, f ) for all large enough i, we must show that λ i ∈ V and φ χi ∈ U . For the latter, note that as χ i → χ and as ν → φ ν is continuous, we know that φ χi ∈ U for all i larger than some i 2 . For the former, note that, since γ i → γ and f is a continuous function with f (γ) = f (γ) > 0, we have that f (γi) |f (γi)| = λ i µ i converges to 1. Since µ i → µ by assumption, this implies that λ i is in the neighborhood V of µ for all large enough i.