A Uniquness Theorem for Twisted Groupoid C*-Algebras

A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS

BECKY ARMSTRONG

Abstract. We present a uniqueness theorem for the reduced C*-algebra of a twist E over a Hausdorff étalegroupoid G. We show that the interior IE of the isotropy of E is a twist over the interior IG of the isotropy of G, and that the reduced twisted groupoid C*- ∗ G E ∗ algebra Cr (I ; I ) embeds in Cr (G; E). We also investigate the full and reduced twisted C*-algebras of the isotropy groups of G, and we provide a sufficient condition under which states of (not necessarily unital) C*-algebras have unique state extensions. We use these results to prove our uniqueness theorem, which states that a C*-homomorphism ∗ ∗ G E of Cr (G; E) is injective if and only if its restriction to Cr (I ; I ) is injective. We also ∗ show that if G is effective, then Cr (G; E) is simple if and only if G is minimal.

1. Introduction 1.1. Background. The purpose of this article is prove a uniqueness theorem for the reduced C*-algebra of a twist E over a second-countable locally compact Hausdorff étale groupoid G. The study of twisted groupoid C*-algebras was initiated by Renault [32], who generalised twisted group C*-algebras by constructing full and reduced C*-algebras ∗ ∗ C (G, σ) and Cr (G, σ) from a second-countable locally compact Hausdorff groupoid G and a continuous 2-cocycle σ on G taking values in the complex unit circle T. Renault also realised C∗(G, σ) as a quotient of the C*-algebra associated to the extension of G by T defined by the 2-cocycle σ. This construction was subsequently extended by Kumjian [22] to include C*-algebras over groupoid twists that don’t necessarily arise from continuous 2-cocycles. More recently, Renault [33] showed that every C*-algebra containing a Cartan subalgebra can be realised as a twisted groupoid C*-algebra, thereby providing a C*-algebraic analogue of Feldman–Moore theory [18, 19, 20]. Renault’s reconstruction theorem is of particular importance to the classification program for C*-algebras, given Li’s re- cent article [29] showing that every simple classifiable C*-algebra has a Cartan subalgebra (and is therefore a twisted groupoid C*-algebra), and the work of Barlak and Li [7,8] describing the connections between the UCT problem and Cartan subalgebras in C*-algebras. The increasing interest in twisted groupoid C*-algebras (see, for instance, [3,6, 10, 11, 15, 16, 17, 23]) has also recently inspired the introduction of twisted Stein- arXiv:2103.03063v2 [math.OA] 7 Sep 2021 berg algebras, which are a purely algebraic analogue of twisted groupoid C*-algebras (see [4,5]). Examples of twisted groupoid C*-algebras include the twisted C*-algebras associated to higher-rank graphs introduced by Kumjian, Pask, and Sims [24, 25, 26, 27], and the more general class of twisted C*-algebras associated to topological higher-rank graphs introduced in the author’s PhD thesis [2].

Date: 8th September 2021. 2020 Mathematics Subject Classiﬁcation. 46L05. Key words and phrases. C*-algebra, groupoid, twist, uniqueness, isotropy. The author would like to thank Nathan Brownlowe and Aidan Sims for many helpful discussions, and for their assistance with preparing and editing this article. The author was supported by an Australian Government Research Training Program Stipend Scholarship.

1 2 BECKY ARMSTRONG

In this article we prove a uniqueness theorem (Theorem 6.4) for the reduced C*-algebra of a twist E over a second-countable locally compact Hausdorff étalegroupoid G. In particular, we show that the interior IE of the isotropy of E is a twist over the interior IG of the ∗ isotropy of G, and that a C*-homomorphism of the reduced twisted C*-algebra Cr (G; E) ∗ G E is injective if and only if its restriction to Cr (I ; I ) is injective. This is an extension of the analogous result [12, Theorem 3.1(b)] for non-twisted groupoid C*-algebras, and also of the result [2, Theorem 5.3.14] appearing in the author’s PhD thesis for twisted groupoid C*-algebras arising from continuous 2-cocycles on groupoids. Although many of the arguments used in this article are inspired by their non-twisted counterparts, the twisted setting differs significantly enough from the non-twisted setting to warrant independent treatment. In particular, although G is an étalegroupoid, the twist E is not an étalegroupoid, and this leads to increased technical complexity in many of our proofs. ∗ One interesting corollary of our main theorem is that if G is effective, then Cr (G; E) is simple if and only if G is minimal (see Corollary 6.10).

1.2. Outline. This article is organised as follows. In Section 2 we establish the relevant background and notation, and we recall various well known useful results relating to twists and twisted groupoid C*-algebras. In particular, in Section 2.1 we recall the definition of a twist E over a Hausdorff étalegroupoid G, and we show that the interior IE of the isotropy of E is a twist over the interior IG of the isotropy of G. In Section 2.2 we recall Kumjian’s construction of the full and reduced twisted groupoid C*-algebras C∗(G; E) ∗ and Cr (G; E), and in Proposition 2.14 we describe the relationship between these C*- algebras and Renault’s twisted groupoid C*-algebras arising from continuous 2-cocycles. This result can be used to translate the results of Sections 4 and6 to the analogous results pertaining to twisted groupoid C*-algebras arising from continuous 2-cocycles that appear in the author’s PhD thesis [2] (see Remark 2.15). Throughout Section 6 we regularly work with twisted C*-algebras associated to the isotropy groups of G (which are discrete, since G is étale),and so in Section 3 we restrict our attention to twisted C*-algebras associated to discrete groups in order to establish the necessary preliminaries. In particular, we recall the universal property of the full twisted group C*-algebra C∗(G, σ) associated to a discrete group G and a 2-cocycle σ on G, and in Theorem 3.2 we translate this universal property to the language of the associated central extension of G by T. In Section 4 we show that the full and reduced twisted C*-algebras of the isotropy groups of IG are quotients of the full and reduced twisted C*-algebras of the groupoid IG itself (see Theorem 4.3). For the full C*-algebra, we quotient C∗(IG; IE ) by the ideal generated by functions that vanish on the given isotropy group, but surprisingly, it turns out that this is not the correct ideal to quotient by in the reduced setting. Although the discovery of this fact did not cause us any problems when proving our main theorem, it was somewhat unexpected (at least to the author), and so we provide a proof using an example due to Willett [38] of a nonamenable groupoid whose full and reduced C*-algebras coincide (see Theorem 4.10). A substantial portion of the author’s PhD thesis is dedicated to extending well known results of Anderson [1] about states of unital C*-algebras to the non-unital setting (see [2, Section 5.2]). We reproduce this material in Section 5 of the article, as we have been unable to find explicit proofs of these results in the literature, despite them apparently being well known (for instance, they are used in [12]). We apply these results to twisted groupoid C*-algebras in Section 6. The main result of Section 5 is Theorem 5.3, in which we provide a sufficient compressibility condition under which states of (not necessarily unital) C*-algebras have unique state extensions. A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 3

∗ G E ∗ In Section 6 we show that there is an embedding ιr of Cr (I ; I ) into Cr (G; E), and G ∗ ∗ G E that if I is closed, then there is a conditional expectation from Cr (G; E) to Cr (I ; I ) extending restriction of functions. We also show that these results hold for the full C*- algebras when IG is amenable. We then present our main theorem (Theorem 6.4), which ∗ states that a C*-homomorphism Ψ of Cr (G; E) is injective if and only if the homomorphism ∗ G E Ψ◦ιr of Cr (I ; I ) is injective. We use this theorem to prove Corollary 6.10, which states ∗ that if G is effective, then Cr (G; E) is simple if and only if G is minimal. The uniqueness theorem also has potential applications to the study of the ideal structure of twisted C*- algebras associated to Hausdorff étalegroupoids, and in fact has already been used by the author, Brownlowe, and Sims in [3] to characterise simplicity of twisted C*-algebras associated to Deaconu–Renault groupoids.

2. Preliminaries In this section we present the necessary background on twists over Hausdorff étale groupoids and the associated (full and reduced) twisted groupoid C*-algebras. Although groupoid C*-algebras were introduced by Renault in [32], we will frequently reference Sims’ treatise [35] on Hausdorff étalegroupoids and their C*-algebras instead, as it aligns more closely with our setting. The results in this section are presumably well known, but we have presented proofs wherever we have been unable to find them in the literature, or whenever we have felt the need to expand on the level of detail given in existing literature. We begin by recalling some preliminaries on groupoids from [35, Chapter 8]. Throughout this article, G will denote a second-countable locally compact Hausdorff groupoid with unit space G(0), which is étale in the sense that the range and source maps r, s: G → G(0) are local homeomorphisms. We refer to such a groupoid as a Hausdorff étalegroupoid, and we denote the set of composable pairs in G by G(2). If G is étale,then G(0) is an open subset of G, and the range, source, and multiplication maps are all open. We call a subset B of G a bisection if there is an open subset U of G containing B such (0) that r|U and s|U are homeomorphisms onto open subsets of G . Every Hausdorff étale groupoid has a (countable) basis of open bisections. Given subsets A, B ⊆ G, we write AB := {αβ :(α, β) ∈ (A × B) ∩ G(2)} and A−1 := {α−1 : α ∈ A}, and for γ ∈ G, we write (0) −1 u −1 γA := {γ}A and Aγ := A{γ}. Given u ∈ G , we define Gu := s (u), G := r (u), and u u (0) u u Gu := Gu ∩ G . For each u ∈ G , the relative topology on Gu, G , and Gu is discrete, and u Gu is a countable closed subgroup of G, called an isotropy group. The isotropy subgroupoid u G of G is the groupoid Iso(G) := ∪u∈G(0) Gu = {γ ∈ G : r(γ) = s(γ)}. We write I for the topological interior of Iso(G), and we note that if G is a Hausdorff étalegroupoid, then so G (0) G (0) (0) G G u is I . Since G is open in G, the unit space of I is G . For each u ∈ G , Iu = I ∩Gu is an isotropy group of IG. We say that G is effective if IG = G(0). We call a subset U of G(0) invariant if s(γ) ∈ U =⇒ r(γ) ∈ U for all γ ∈ G, and we say that G is minimal if G(0) has no nonempty proper open (or, equivalently, closed) invariant subsets.

2.1. Twists over Hausdorff étalegroupoids. Groupoid twists and their associated C*-algebras were introduced by Kumjian [22] and subsequently studied by Renault [33]; however, for consistency of terminology and notation, we will continue to reference Sims’ treatise [35]. We begin by recalling the definition of a twist from [35, Definition 11.1.1].

Definition 2.1. A twist (E, i, q) over a Hausdorff étalegroupoid G is a sequence

q (0) i G × T ,→ E G, 4 BECKY ARMSTRONG where the groupoid G(0) × T is viewed as a trivial group bundle with ﬁbres T, E is a locally compact Hausdorﬀ groupoid with unit space E (0) = iG(0) ×{1}, and the following additional conditions hold. (a) The maps i and q are continuous groupoid homomorphisms that restrict to home- (0) (0) omorphisms of unit spaces, and we identify E with G via q|E(0) . −1 (0) (b) The sequence is exact, in the sense that i {x} × T = q (x) for each x ∈ G , i is injective, and q is surjective. (c) The groupoid E is a locally trivial G-bundle, in the sense that for each α ∈ G, there is an open neighbourhood Uα ⊆ G of α, and a continuous map Pα : Uα → E such that

(i) q ◦ Pα = idUα ; and

(ii) the map φPα :(β, z) 7→ i(r(β), z) Pα(β) is a homeomorphism from Uα ×T onto −1 q (Uα). (d) The image of i is central in E, in the sense that i(r(ε), z) ε = ε i(s(ε), z) for all ε ∈ E and z ∈ T. We sometimes denote a twist (E, i, q) over G simply by E. We call a continuous map Pα : Uα → E satisfying condition (c)(i)a (continuous) local section for q, and we call a collection (Uα,Pα, φPα )α∈G satisfying condition (c)a local trivialisation of E. If G is a discrete group, then a twist over G as defined above is a central extension of G. It is well known (see, for instance, [14, Theorem IV.3.12]) that there is a one-to-one correspondence between central extensions of a discrete group and 2-cocycles on the group. This result does not hold in general for groupoids; however, every continuous T-valued 2-cocycle on a groupoid G does give rise to a twist over G, as we show in Example 2.3. To make sense of this example, we first recall the definition of a groupoid 2-cocycle. Definition 2.2. A continuous T-valued2 -cocycle on a topological groupoid G is a continuous map σ : G(2) → T satisfying (i) σ(α, β) σ(αβ, γ) = σ(α, βγ) σ(β, γ), for all α, β, γ ∈ G such that s(α) = r(β) and s(β) = r(γ); and (ii) σ(r(γ), γ) = σ(γ, s(γ)) = 1, for all γ ∈ G. Example 2.3. Let G be a Hausdorff étalegroupoid and let σ : G(2) → T be a continuous 2-cocycle. Let Eσ = G ×σ T be the set G × T endowed with the product topology. The formulae (α, w)(β, z) := αβ, σ(α, β)wz and (α, w)−1 := α−1, σ(α, α−1) w define multiplication and inversion operations on Eσ, under which Eσ is a locally compact (0) Hausdorff groupoid. Let iσ : G × T → Eσ be the inclusion map and let qσ : Eσ → G be the projection onto the first coordinate. Then (Eσ, iσ, qσ) is a twist over G. A routine argument shows that if (E, i, q) is a twist over a Hausdorff étale groupoid, then the formulae z · ε := i(r(ε), z) ε and ε · z := ε i(s(ε), z) define continuous free left and right actions of T on E. Centrality of the image of i implies that z·ε = ε·z for all z ∈ T and ε ∈ E. This action has the following additional properties. Lemma 2.4. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. (a) For each fixed z ∈ T, the map ϕz : ε 7→ z · ε is a homeomorphism of E. (b) If ε, ζ ∈ E satisfy q(ε) = q(ζ), then there is a unique z ∈ T such that ε = z · ζ.

(c) Let (Uα,Pα, φPα )α∈G be a local trivialisation of (E, i, q). There is a unique continuous map t : q−1(U ) → such that φ−1(ε) = (q(ε), t (ε)) for all ε ∈ q−1(U ). α α T Pα α α A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 5

Proof. For part (a), ﬁx z ∈ T. Since the action of T on E is continuous, ϕz is a continuous bijection with inverse ϕz, and hence ϕz is a homeomorphism. Part (b) is [35, Lemma 11.1.3]. −1 −1 For part (c), ﬁx ε ∈ q (Uα). Since φPα : Uα × T → q (Uα) is a homeomorphism, there is a unique pair (βε, zε) ∈ Uα × T such that

ε = φPα (βε, zε) = i(r(βε), zε) Pα(βε) = zε · Pα(βε). (2.1) (0) −1 (0) Since i(G × T) = q (G ) and q ◦ Pα = idUα , it follows from equation (2.1) that −1 q(ε) = βε. Since zε is unique, there is a map tα : q (Uα) → T given by tα(ε) := zε, and equation (2.1) implies that φ−1(ε) = (q(ε), t (ε)). Letting π : G × → denote the Pα α 2 T T projection map, we see that t = π ◦ φ−1, and so t is continuous because it is composed α 2 Pα α of continuous maps. We now show that the continuous local sections of Definition 2.1(c)(i) can always be chosen to be defined on bisections of G, and to map units of G to the corresponding units of E. Lemma 2.5. Every twist (E, i, q) over a Hausdorff étalegroupoid G has a local trivialisa- (0) (0) tion (Bα,Pα, φPα )α∈G such that for all α ∈ G, Bα is a bisection and Pα(Bα ∩ G ) ⊆ E .

Proof. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G, and let (Uα,Sα, ψSα )α∈G be a local trivialisation of (E, i, q). For each α ∈ G, let Dα be an open bisection of G such that α ∈ Dα ⊆ Uα, and define ( D ∩ G(0) if α ∈ G(0) : α Bα = (0) (0) Dα\G if α∈ / G .

(0) Since G is a Hausdorff étalegroupoid, G is clopen, and hence each Bα is an open bisection of G containing α. There are now two cases to deal with. For the first case, fix α ∈ G\G(0). Define (0) (0) Pα := Sα|Bα and φPα := ψSα |Bα×T. Then Pα(Bα ∩ G ) = Sα(∅) ⊆ E . Since Sα is a continuous local section for q, so is Pα. Since Bα is an open subset of Uα and −1 ψSα (Bα × T) = q (Bα), it follows that φPα is a homeomorphism from Bα × T onto −1 q (Bα) satisfying φPα (β, z) = i(r(β), z) Pα(β) for all (β, z) ∈ Bα × T. (0) (0) For the second case, fix x ∈ G . Then Bx ⊆ G . Define Px : Bx → E by Px(y) := −1 (q|E(0) ) (y). Then Px is continuous because q|E(0) a homeomorphism and the inclusion (0) map E ,→ E is continuous. It is clear that q ◦ Px = idBx , and so Px is a continuous (0) −1 (0) local section for q satisfying Px(Bx ∩ G ) = (q|E(0) ) (Bx) ⊆ E . By Lemma 2.4(c), there is a unique continuous map t : q−1(U ) → such that ψ−1(ε) = (q(ε), t (ε)) for all x x T Sx x −1 ε ∈ q (Ux). In particular, for all y ∈ Bx, we have ψ−1P (y) = q(P (y)), t (P (y)) = y, t (P (y)). (2.2) Sx x x x x x x Define fx : Bx ×T → Bx ×T by fx(y, z) := y, z tx(Px(y)) . Since tx and Px are continuous, −1 fx is a homeomorphism with inverse fx :(y, z) 7→ y, z tx(Px(y)) . Define φPx := ψSx ◦fx. −1 Then φPx is a homeomorphism from Bx × T onto q (Bx). Fix (y, z) ∈ Bx × T. Using that i is a homomorphism for the third equality and equation (2.2) for the final equality, we see that φPx (y, z) = ψSx y, z tx(Px(y)) = i y, z tx(Px(y)) Sx(y) = i(y, z) i y, tx(Px(y)) Sx(y) = i(y, z) ψSx y, tx(Px(y)) = i(y, z) Px(y). 6 BECKY ARMSTRONG

Thus we have constructed a local trivialisation (Bα,Pα, φPα )α∈G for (E, i, q) with the de- sired properties. Remark 2.6. In some texts (for instance, [10, Definition 3.1]), the existence of continuous local sections (and the induced local trivialisation) is omitted from the definition of a twist (E, i, q), and instead, i is defined to be a homeomorphism onto the open set q−1(G(0)), and q is defined to be a continuous open surjection. These conditions imply that q admits continuous local sections (see [10, Proposition 3.4]), and since i has a continuous inverse defined on q−1(G(0)), an argument similar to the one used in the proof of [5, Proposition 4.8(c)] shows that these local sections induce a local trivialisation of the twist. Hence the twists of [10, Definition 3.1] are twists in the sense of Definition 2.1. On the other hand, in Lemma 2.7 we show that, given a twist (E, i, q) in the sense of Definition 2.1, the map i is a homeomorphism onto the open set q−1(G(0)), and the map q is a continuous open surjection. Hence Definition 2.1 is in fact equivalent to [10, Definition 3.1]. Lemma 2.7. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. (a) The map q is a continuous open surjection, and G has the quotient topology. (b) The range, source, and multiplication maps on E are all open. (c) The map i is a homeomorphism onto q−1(G(0)), which is an open subset of E. Proof. For part (a), note that q is a continuous surjection by Definition 2.1. We first show that G has the quotient topology. Let X be a subset of G. If X is open, then q−1(X) is open in E, because q is continuous. Suppose instead that q−1(X) is open in E. We must show that X is open in G. Choose a local trivialisation (Uα,Pα, φPα )α∈G of (E, i, q). Fix −1 −1 −1 α ∈ X. The set q (X) ∩ q (Uα) is an open subset of q (Uα) that is closed under the action of on E, and hence its (open) image under φ−1 is of the form V × , for some T Pα α T open subset Vα of Uα. We have −1 −1 Vα = q φPα (Vα × T) = q q (X) ∩ q (Uα) = X ∩ Uα ⊆ X, and so Vα is an open neighbourhood of α contained in X. Hence X is open in G, and q is a quotient map. We now show that q is an open map. Let Y be an open subset of E. We must show that q(Y ) is open. Since G has the quotient topology, q(Y ) is open in G if and only if q−1(q(Y )) is open in E. Recall from Lemma 2.4(a) that for each z ∈ T, the map ϕz : ε 7→ z · ε is a −1 homeomorphism of E, and so ϕz(Y ) is open. Since ε ∈ q (q(Y )) if and only if ε = z ·ζ for −1 −1 some z ∈ T and ζ ∈ Y , we have q (q(Y )) = T · Y = ∪z∈T ϕz(Y ). Therefore, q (q(Y )) is open in E because it is a union of open sets. (0) For part (b), first note that the range map rG : G → G is open, because G is étale. Since q : E → G is a groupoid homomorphism that restricts to a homeomorphism of unit −1 (0) spaces, we have rE = (q|E(0) ) ◦ rG ◦ q, and hence part (a) implies that rE : E → E is an open map. Composing the range map with the inverse map gives the source map, and so (0) since inversion is a homeomorphism, the source map sE : E → E is also open. Therefore, by [35, Lemma 8.4.11], the multiplication map on E is open. For part (c), first note that i(G(0) × T) = q−1(G(0)) is an open subset of E, because q is continuous and G(0) is open in G. We now show that i is an open map. Let U be an open subset of G(0) and let W be an open subset of T. By Lemma 2.5, we can find a local trivialisation (Bα,Pα, φPα )α∈G of (E, i, q) such that for all α ∈ G, Bα is a bisection of G (0) (0) (0) and Pα(Bα ∩ G ) ⊆ E . For each x ∈ U, define Dx := Bx ∩ U, so that Dx ⊆ G and S (0) U = x∈U Dx. Fix x ∈ U. Since Px(Dx) ⊆ E , we have

φPx (Dx × W ) = W · Px(Dx) = i(Dx × W ). A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 7

−1 By Definition 2.1(c)(ii), φP is a homeomorphism onto the open set q (Bx), and so since x S Dx × W is open, it follows that i(Dx × W ) is open. Hence i(U × W ) = x∈U i(Dx × W ) is an open subset of E, and so i is an open map. Definition 2.8. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Any map P : G → E satisfying q ◦ P = idG is called a (global) section for q. The following result shows that there is a one-to-one correspondence between continuous 2-cocycles on a Hausdorff étalegroupoid G and twists over G admitting continuous global sections. Note that in general, a twist over a Hausdorff étale groupoid need not admit any continuous global sections. Proposition 2.9. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose that P : G → E is a continuous global section for q. Then there is a continuous 2-cocycle (2) (2) σ : G → T such that P (α) P (β) = σ(α, β)·P (αβ) for all (α, β) ∈ G . Let (Eσ, iσ, qσ) be the twist defined in Example 2.3. The map φP : Eσ → E given by φP (γ, z) := z·P (γ) defines an isomorphism of twists, in the sense that φP is a topological groupoid isomorphism that makes the diagram

Eσ i σ qσ

(0) G × T φP G. i q E commute. Moreover, there is a continuous global section S : G → E for q satisfying S(G(0)) ⊆ E (0). Proof. By [22, Section 4, Fact 1], every continuous global section for q induces a continuous 2-cocycle satisfying the given formula. It is observed in [22, Section 4, Remark 2] that the map φP :(γ, z) 7→ z · P (γ) defines an isomorphism of the twists (Eσ, iσ, qσ) and (E, i, q). (Alternatively, see the proof of the analogous result [5, Proposition 4.8] for discrete twists, which holds in our non-discrete setting.) To see that continuous global sections can be chosen to map units to units, observe that (G, P, φP )α∈G is a local trivialisation of E, and so we can apply the argument of Lemma 2.5 (without replacing G by bisections in the local trivialisation). The following result and the subsequent corollary will be frequently used throughout the remainder of the article, without necessarily being explicitly referenced. These results allow us to restrict our attention to twists over the unit space, the interior of the isotropy, or the isotropy groups of a Hausdorff étalegroupoid. Proposition 2.10. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose −1 : : (0) that H is an open or closed étalesubgroupoid of G. Define EH = q (H), iH = i|H ×T, and qH := q|EH . Then H is a Hausdorff étalegroupoid, and (EH, iH, qH) is a twist over H. Proof. Since H is an open or closed subgroupoid of G, it follows that H is a locally −1 compact Hausdorff space. Since q is continuous, EH = q (H) is open or closed in E, and hence EH is a locally compact Hausdorff subspace of E. To see that EH is a subgroupoid −1 of E, fix ε, ζ ∈ EH such that s(ε) = r(ζ). We must show that εζ, ε ∈ EH. Since q is a groupoid homomorphism, we have s(q(ε)) = q(s(ε)) = q(r(ζ)) = r(q(ζ)), and (2) so q(ε), q(ζ) ∈ H . Hence q(εζ) = q(ε)q(ζ) ∈ H, and so εζ ∈ EH. We also have −1 −1 −1 q(ε ) = q(ε) ∈ H, and so ε ∈ EH. Thus EH is a subgroupoid of E with unit space (0) (0) −1 (0) (0) EH = E ∩q (H ) = iH H ×{1} . Since H and EH are open or closed subgroupoids of G and E, respectively, conditions (a), (b), and (d) of Definition 2.1 follow easily from 8 BECKY ARMSTRONG the fact that (E, i, q) is a twist over G. To see that condition (c) is satisfied, choose a local trivialisation (Uα,Pα, φP )α∈G of (E, i, q), and define Dα := Uα ∩ H for each α ∈ H. α A routine argument then shows that Dα,Pα|Dα , φPα |Dα×T α∈H is a local trivialisation of (EH, iH, qH). Therefore, (EH, iH, qH) is a twist over H. Corollary 2.11. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. (a) The groupoid q−1(G(0)) is a twist over G(0). (b) The isotropy subgroupoid IE of E is a twist over the isotropy subgroupoid IG of G. (0) E G (c) For each u ∈ E , the isotropy group Iu is a twist over the isotropy group Iq(u). Proof. Part (a) follows immediately from Proposition 2.10 since G(0) is an open étale subgroupoid of G. For part (b), we will show that q−1(IG) = IE , because the result then follows immediately from Proposition 2.10 since IG is an open étalesubgroupoid of G. We first show that q−1Iso(G) = Iso(E). For this, fix ε ∈ E. Since q : E → G is a groupoid homomorphism that restricts to a homeomorphism of unit spaces, we have r(q(ε)) = s(q(ε)) ⇐⇒ q(r(ε)) = q(s(ε)) ⇐⇒ r(ε) = s(ε), (2.3) and so q(ε) ∈ Iso(G) if and only if ε ∈ Iso(E). Thus q−1Iso(G) = Iso(E), as claimed. Since IG is open and q is continuous, q−1(IG) is an open subset of E contained in Iso(E), and so q−1(IG) ⊆ IE . Since IE is open and q is an open map by Lemma 2.7(a), q(IE ) is an open subset of G contained in Iso(G), and hence IE ⊆ q−1(IG). Therefore, q−1(IG) = IE . (0) −1 G E For part (c), fix u ∈ E . The proof of part (b) implies that q Iq(u) = Iu , and so the G G result follows from Proposition 2.10 since Iq(u) is a discrete closed subgroupoid of I . 2.2. Twisted groupoid C*-algebras. In this section we recall Kumjian’s construction (given in [22]) of the full and reduced twisted groupoid C*-algebras associated to a twist over a Hausdorff étalegroupoid. We also recall Renault’s construction (given in [32]) of the full and reduced twisted groupoid C*-algebras arising from a continuous 2-cocycle on a Hausdorff étalegroupoid, and we describe the relationship between these two construc- tions in Proposition 2.14. Given a locally compact Hausdorff space X, we write C(X) for the vector space of continuous complex-valued functions on X under pointwise operations. For each f ∈ C(X), we define osupp(f) := f −1(C\{0}), and we write supp(f) for the closure of supp(f) in X. We define Cc(X) := {f ∈ C(X) : supp(f) is compact}, and we write C0(X) for the collection of continuous functions on X vanishing at infinity, which is the completion of the subspace Cc(X) with respect to the uniform norm k·k∞. Suppose that (E, i, q) is a twist over a Hausdorff étalegroupoid G. For each γ ∈ G, the set q−1(γ) is homeomorphic to T, since E is a locally trivial G-bundle. Since the Haar measure on T is rotation-invariant, pulling it back to q−1(γ) gives a measure that is −1 (0) independent of our choice of ε ∈ q (γ). For each u ∈ E , we endow Eu with a measure −1 λu that agrees with these pulled back copies of Haar measure on q (γ) for each γ ∈ Gq(u), and so each q−1(γ) has measure 1. We define a measure λu on each E u in a similar fashion. We say that a function f : E → C is T-equivariant if f(z · ε) = z f(ε) for all z ∈ T and ε ∈ E. The collection

Σc(G; E) := {f ∈ Cc(E): f is T-equivariant} is a ∗-algebra under pointwise addition and scalar multiplication, multiplication given by the convolution formula Z Z −1 −1 r(ε) (fg)(ε) := f(εζ ) g(ζ) dλs(ε)(ζ) = f(η) g(η ε) dλ (η), r(ε) Es(ε) E A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 9 and involution given by f ∗(ε) := f(ε−1). Taking P : G → E to be any (not necessarily continuous) global section for q, it follows from the T-equivariance of f, g ∈ Σc(G; E) that for all ε ∈ E, X X (fg)(ε) = fεP (α)−1 gP (α) = fP (β) gP (β)−1ε. (2.4)

α∈Gq(s(ε)) β∈Gq(r(ε))

Although Cc(G) is spanned by functions supported on open bisections of G (see, for instance, [35, Lemma 9.1.3]), this is not the case for Σc(G; E), because E is not étale. However, we do have the following similar result. Lemma 2.12. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Then

Σc(G; E) = span{f ∈ Σc(G; E): q(supp(f)) is a bisection of G}.

Proof. Fix f ∈ Σc(G; E). For each γ ∈ q(supp(f)), let Bγ be an open bisection of G containing γ. Since q(supp(f)) is compact, there is a ﬁnite subset F of q(supp(f)) such that q(supp(f)) ⊆ ∪γ∈F Bγ. As in [28, Remark 2.9], let {hγ : γ ∈ F } be a partition of unity subordinate to {Bγ ∩ q(supp(f)) : γ ∈ F }. For each γ ∈ F , the function fγ : ε 7→ f(ε) hγ(q(ε)) belongs to Cc(E), and q(supp(fγ)) ⊆ supp(hγ) ⊆ Bγ. Fix γ ∈ F . The T-equivariance of f implies that for all z ∈ T and ε ∈ E,

fγ(z · ε) = f(z · ε) hγ(q(z · ε)) = z f(ε) hγ(q(ε)) = z fγ(ε), P P and hence fγ ∈ Σc(G; E). Since γ∈F hγ = 1, we have f = γ∈F fγ, and hence Σc(G; E) ⊆ span f ∈ Σc(G; E): q(supp(f)) is a bisection of G . The other containment is clear, and so this completes the proof. The full twisted groupoid C*-algebra associated to the pair (G, E) is deﬁned to be the ∗ completion C (G; E) of Σc(G; E) with respect to the full norm

kfk := sup{kπ(f)k : π is a ∗-representation of Σc(G; E)} . (0) For each unit u ∈ E , there is a ∗-representation πu of Σc(G; E) on the Hilbert space 2 L (Gq(u); Eu) consisting of square-integrable T-equivariant functions on Eu, which is given by extension of the convolution formula. We call each πu the regular representation of I G E Σc(G; E) associated to u, and we write πu for the regular representation of Σc(I ; I ) ∗ associated to u. The reduced twisted groupoid C*-algebra Cr (G; E) is deﬁned to be the completion of Σc(G; E) with respect to the reduced norm (0) kfkr := sup kπu(f)k : u ∈ E .

For all f ∈ Σc(G; E), we have kfk∞ ≤ kfkr ≤ kfk, with equality throughout if q(supp(f)) is a bisection of G. If G is amenable, then the full and reduced norms agree on Σc(G; E). We deﬁne (0) (0) D0 := f ∈ Σc(G; E): q(supp(f)) ⊆ G = f ∈ Σc(G; E) : supp(f) ⊆ i(G × T) , (0) ∼ (0) and note that there is a ∗-isomorphism Cc(G ) = D0 mapping h ∈ Cc(G ) to the function fh : i(x, z) 7→ z h(x) (see [35, Lemma 11.1.9]). This ∗-isomorphism extends to an (0) ∗ isomorphism of C0(G ) to the completion Dr of D0 in Cr (G; E). There is a faithful con- ∗ ditional expectation Φr : Cr (G; E) → Dr that extends restriction of functions in Σc(G; E) to the set q−1(G(0)) = i(G(0) × T) (see [33, Proposition 4.3] and [35, Proposition 11.1.13]). I ∗ G E We write Φr for the corresponding conditional expectation from Cr (I ; I ) to Dr. There ∗ is also a conditional expectation from C (G; E) to the completion of D0 in the full norm −1 (0) that extends restriction of functions from Σc(G; E) to q (G ), but this conditional expectation is not necessarily faithful. 10 BECKY ARMSTRONG

We now show that every ∗-homomorphism of Σc(G; E) into a C*-algebra A extends uniquely to a C*-homomorphism of C∗(G; E) into A. This is an extension of the well known result [2, Lemma 3.3.22] to the setting of C*-algebras of groupoid twists. Lemma 2.13. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose that A is a C*-algebra and that φ:Σc(G; E) → A is a ∗-homomorphism. Then φ extends uniquely to a C*-homomorphism φ: C∗(G; E) → A satisfying kφk = kφk. Proof. The result follows by a similar argument to the proof of [2, Lemma 3.3.22]. We now recall Renault’s construction of twisted groupoid C*-algebras arising from continuous groupoid 2-cocycles. For our purposes, it suffices to consider a Hausdorff étale groupoid G, although we note that Renault deals with groupoids that are not necessarily (2) étalein [32]. Suppose that σ : G → T is a continuous 2-cocycle. We write Cc(G, σ) for the ∗-algebra consisting of continuous, compactly supported, complex-valued functions on G equipped with pointwise addition and scalar multiplication, multiplication given by the twisted convolution formula X X (fg)(γ) := σ(α, β) f(α) g(β) = σ(γη−1, η) f(γη−1) g(η),

(α,β)∈G(2), η∈Gs(γ) αβ=γ

∗ −1 −1 and involution given by f (γ) := σ(γ, γ ) f(γ ). The full and reduced norms on Cc(G, σ) are defined in a similar fashion to the full and reduced norms on Σc(G; E), and by complet- ing Cc(G, σ) with respect to these norms, we obtain the full and reduced twisted groupoid ∗ ∗ C*-algebras C (G, σ) and Cr (G, σ), respectively. In Section 2.1 we showed that every continuous 2-cocycle σ on G gives rise to a twist Eσ over G by T, and in Proposition 2.9 we showed that every twist E admitting a continuous ∼ global section gives rise to a continuous 2-cocycle σ such that Eσ = E. If φ: E1 → E2 is an isomorphism of twists over G, then a routine argument shows that the map f 7→ f ◦ φ is an isomorphism from Σc(G; E2) to Σc(G; E1). Thus we can use Proposition 2.9 to describe the relationship between twisted groupoid C*-algebras arising from continuous 2-cocycles, and those arising from twists admitting continuous global sections, as follows. Proposition 2.14. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose that P : G → E is a continuous global section for q. Let σ : G(2) → T be the continuous 2-cocycle induced by P , as described in Proposition 2.9. Let σ denote the continuous 2-cocycle obtained by composing σ with the complex conjugation map on T. There is a ∗-isomorphism ψP :Σc(G; Σ) → Cc(G, σ) given by ψP (f) := f ◦ P . This isomorphism ∗ ∼ ∗ ∗ ∼ ∗ extends to C*-isomorphisms C (G; E) = C (G, σ) and Cr (G; E) = Cr (G, σ). ∼ Proof. Let (Eσ, iσ, qσ) be the twist defined in Example 2.3. Since E = Eσ by Proposi- tion 2.9, and since (γ, 1) 7→ γ is a continuous global section for qσ, the result follows from the “n = −1” case of [11, Lemma 3.1(b)]. Remark 2.15. Using Proposition 2.14, the results in Sections 4 and6 can be translated into analogous results for twisted groupoid C*-algebras arising from continuous 2-cocycles. Alternatively, we refer the reader to [2, Chapter 5] for the corresponding results written explicitly in this framework.

3. A universal property for twisted group C*-algebras In this section, we describe twisted C*-algebras of countable discrete groups. Since every such group is a Hausdorff étalegroupoid, the preliminaries given in Section 2 are all applicable here. In particular, since every twist over a discrete group admits a (trivially A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 11 continuous) global section, Proposition 2.9 applies to twists over countable discrete groups. Thus, a twisted group C*-algebra can be viewed as arising either from a continuous T- valued 2-cocycle on the group, or from a central extension of the group by T. The main purpose of this section is to translate the universal property for the full twisted C*-algebra C∗(G, σ) associated to a countable discrete group G and a T-valued 2-cocycle σ on G into the language of the associated central extension G ×σ T of G by T (see Theorem 3.2). This result is presumably well known, but we were unable to find a proof of it in the literature. Suppose that G is a countable discrete group with identity e, and that σ : G×G → T is a 2-cocycle. Given a C*-algebra A with identity 1A, we say that u: G → A is a σ-twisted unitary representation of G in A if each ug is a unitary element of A, and uguh = σ(g, h) ugh ∗ −1 for all g, h ∈ G. This implies that ue = 1A, and ug = σ(g, g ) ug−1 for each g ∈ G. The ∗ map δ : G → C (G, σ) sending each g ∈ G to the point-mass function δg at g is a σ-twisted ∗ ∗ unitary representation of G such that C (G, σ) = C {δg : g ∈ G} , and the following universal property holds: if B is a unital C*-algebra and u: G → B is a σ-twisted unitary ∗ representation of G in B, then there is a homomorphism λu : C (G, σ) → B such that λu(δg) = ug for each g ∈ G. Proposition 3.1. Let (E, i, q) be a twist over a countable discrete group G. Fix ε ∈ E, T and define δε : E → C by ( w if ζ = w · ε for some w ∈ T T δε (ζ) := 0 if ζ 6= w · ε for all w ∈ T.

T ∗ ∗ Then Σc(G; E) = span δε : ε ∈ E , and Cr (G; E) and C (G; E) are both unital with T T identity δe , where e is the identity of E. For all ε, ζ ∈ E and z ∈ T, δε is a unitary T ∗ T T T T T T element of Σc(G; E) satisfying (δε ) = δε−1 , and we have δz·ε = z δε and δε δζ = δεζ .

T Proof. Fix ε ∈ E. We first prove that δε ∈ Σc(G; E). Let P : E → G be a section for q, and let φP : G × T → E be the map (γ, z) 7→ z · P (γ), which is a homeomorphism by Proposition 2.9. For each ζ ∈ E, let zζ be the unique element of T such that ζ = φP (q(ζ), zζ ) = zζ · P (q(ζ)). For all ζ ∈ E, we have q(ζ) = q(ε) if and only if ζ = w · ε for T −1 a unique w ∈ T, and so osupp δε = {w · ε : w ∈ T} = q (q(ε)). Since T is compact and T the action of T on E is continuous, osupp δε is compact. If ζ = w · ε for some w ∈ T, T then zζ = w zε, and so δε (ζ) = zε zζ . Therefore, since the map ζ 7→ zζ is continuous by T −1 Lemma 2.4(c), we deduce that δε |q−1(q(ε)) is continuous. Thus, since q (q(ε)) is clopen, T T δε is continuous on E. To see that δε is T-equivariant, fix ζ ∈ E and w ∈ T, and note that w · ζ ∈ q−1(q(ε)) if and only if ζ ∈ q−1(q(ε)). Thus, if ζ ∈ q−1(q(ε)), then T T −1 T T T δε (w · ζ) = w zε zζ = w δε (ζ), and if ζ∈ / q (q(ε)), then δε (w · ζ) = 0 = w δε (ζ). So δε is T T-equivariant, and hence δε ∈ Σc(G; E). T To see that Σc(G; E) ⊆ span{δε : ε ∈ E}, fix f ∈ Σc(G; E), and define F := q(supp(f)). P T Then F is a finite subset of G. We claim that f = γ∈F f(P (γ)) δP (γ). Fix ε ∈ E. If −1 T ε∈ / supp(f), then q(ε) 6∈ F , and hence for all γ ∈ F , we have ε∈ / q (γ) = osupp δP (γ) . Suppose that ε ∈ supp(f). Then there exists a unique γ ∈ F such that q(ε) = γ. Hence T ε = φP (γ, zε) = zε · P (γ), and so δP (γ)(ε) = zε. Thus, since f is T-equivariant, we have

T f(ε) = f zε · P (γ) = zε f(P (γ)) = f(P (γ)) δP (γ)(ε). (3.1)

T Since δP (η)(ε) = 0 for all η ∈ F \{γ}, we deduce from equation (3.1) that

X T T f = f(P (γ)) δP (γ) ∈ span{δε : ε ∈ E}. γ∈F 12 BECKY ARMSTRONG

Fix ε, ζ ∈ E and z ∈ T. Then ( z w if ζ = w · ε for some w ∈ T T z δε (ζ) = 0 if ζ 6= w · ε for all w ∈ T ( t if ζ = (tz) · ε for some t ∈ = T 0 if ζ 6= (tz) · ε for all t ∈ T T = δz·ε(ζ),

T T and hence z δε = δz·ε. For all ξ ∈ E, we have Z Z T T T −1 T T −1 T (δε δζ )(ξ) = δε (ξη ) δζ (η) dλ(η) = δε w · (ξζ ) δζ (w · ζ) dw E T Z T −1 T −1 T = w w δε (ξζ ) dw = δε (ξζ ) = δεζ (ξ), T T T T and hence δε δζ = δεζ . T T T T T T Let e be the identity of E. Since Σc(G; E) = span δζ : ζ ∈ E and δe δζ = δζ = δζ δe ∗ ∗ T for all ζ ∈ E, it follows that Cr (G; E) and C (G; E) are both unital with identity δe . For T ∗ T −1 T T ∗ T all ε, η ∈ E, we have (δε ) (η) = δε (η ) = δε−1 (η), and hence (δε ) = δε−1 . It follows that T each δε is a unitary element of Σc(G; E). In the following theorem, we use the universal property of C∗(G, σ) to give a universal property for C∗(G; E). Theorem 3.2. Let (E, i, q) be a twist over a countable discrete group G. Suppose that {tε : ε ∈ E} is a family of unitary elements of a unital C*-algebra A such that tz·ε = z tε ∗ and tε tζ = tεζ for all ε, ζ ∈ E and z ∈ T. Then there is a homomorphism πt : C (G; E) → T T A satisfying πt(δε ) = tε for each ε ∈ E, where δε is deﬁned as in Proposition 3.1. Before proving Theorem 3.2, we need the following preliminary result. Lemma 3.3. Let (E, i, q) be a twist over a countable discrete group G, and let P : G → E be a section for q. Let σ : G×G → T be the 2-cocycle induced by P , as described in Propo- ∗ ∗ sition 2.9, and let ψP : C (G; E) → C (G, σ) be the C*-isomorphism of Proposition 2.14. T For each ε ∈ E, deﬁne δε ∈ Σc(G; E) as in Proposition 3.1, and let zε ∈ T denote the T unique element of T such that ε = zε · P (q(ε)). Then ψP δε = zε δq(ε) for each ε ∈ E, where δq(ε) is the point-mass function at q(ε).

Proof. Fix ε ∈ E and γ ∈ G. Then q(ε) = γ if and only if ε = zε · P (γ), and thus

T T ψP δε (γ) = δε P (γ) ( w if P (γ) = w · ε for some w ∈ = T 0 if P (γ) 6= w · ε for all w ∈ T ( w if ε = w · P (γ) for some w ∈ = T 0 if ε 6= w · P (γ) for all w ∈ T ( z if γ = q(ε) = ε 0 if γ 6= q(ε)

= zε δq(ε)(γ). A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 13

Proof of Theorem 3.2. Let P : G → E be a section for q, and for each ε ∈ E, let zε ∈ T denote the unique element of T such that ε = zε · P (q(ε)). As in Proposition 2.9, let σ : G×G → T be the 2-cocycle induced by P , and let φP : G×σ T → E be the isomorphism of topological groups induced by P . For each γ ∈ G, deﬁne uγ := tP (γ). Then each uγ is a unitary element of A. For all α, β ∈ G, we have P (α) P (β) = σ(α, β) · P (αβ) by Proposition 2.9, and hence

uα uβ = tP (α) tP (β) = tP (α)P (β) = tσ(α,β)·P (αβ) = σ(α, β) tP (αβ) = σ(α, β) uαβ. Thus u: G → A is a σ-twisted representation of G in A, and so by the universal property ∗ ∗ of C (G, σ), there is a homomorphism λu : C (G, σ) → A such that λu(δγ) = uγ for each ∗ ∗ γ ∈ G. Let ψP : C (G; E) → C (G, σ) be the C*-isomorphism of Proposition 2.14, and ∗ deﬁne πt := λu ◦ψP : C (G; E) → A. Then πt is a homomorphism, and Lemma 3.3 implies that for each ε ∈ E, we have

T T πt δε = λu ψP δε = λu zε δq(ε) = zε uq(ε) = zε tP (q(ε)) = tzε·P (q(ε)) = tε.

4. Twisted C*-algebras associated to the interior of the isotropy of a Hausdorff etale´ groupoid In this section we study the relationships between the full and reduced twisted C*- algebras associated to the interior IG of the isotropy of a Hausdorff étalegroupoid G and the full and reduced twisted C*-algebras associated to the isotropy groups of IG. The results in this section are extensions of the results in [2, Section 5.1] to the setting of C*-algebras of groupoid twists. We saw in Corollary 2.11 that, given a twist (E, i, q) over a Hausdorff étalegroupoid G, the interior IE of the isotropy of E is a twist over IG, and for each u ∈ E (0), the isotropy E E G group Iu := {ε ∈ I : r(ε) = s(ε) = u} is a twist over the countable discrete group Iq(u). Thus, recalling the notation defined in Proposition 3.1, we have

G E T E Σc Iq(u); Iu = span{δε : ε ∈ Iu }. (0) 2 Recall from Section 2.2 that for each unit u ∈ E , we write L (Gq(u); Eu) for the Hilbert space consisting of square-integrable T-equivariant functions on Eu. We now construct an 2 orthonormal basis for L (Gq(u); Eu). Proposition 4.1. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0), and let Su : Gq(u) → Eu be a (not necessarily continuous) section for q|Eu . For each ε ∈ Eu, T define δε : Eu → C by ( w if ζ = w · ε for some w ∈ T T δε (ζ) := 0 if ζ 6= w · ε for all w ∈ T.

T 2 T T T Then δε ∈ L (Gq(u); Eu). For all ε, ζ ∈ Eu and z ∈ T, we have δε δζ = δε (ζ) and T T −1 δz·ε = z δε . Given any subset X of Gq(u) such that q (X) is measurable, the collection δT : γ ∈ X is an orthonormal basis for L2(X; q−1(X)). Su(γ)

T 2 Proof. We ﬁrst show that δε ∈ L (Gq(u); Eu) for each ε ∈ Eu. Fix ε ∈ Eu. By an analogous T argument to the one used in the proof of Proposition 3.1, δε is T-equivariant. Moreover, Z Z Z T 2 T 2 2 δε (η) dλu(η) = δε (z · ε) dz = |z| dz = 1, Eu T T T 2 and so δε ∈ L (Gq(u); Eu). 14 BECKY ARMSTRONG

For all ε, ζ ∈ Eu, we have Z Z Z T T T T T T T T δε δζ = δε (η) δζ (η) dλu(η) = δε (z · ζ) δζ (z · ζ) dz = z z δε (ζ) dz = δε (ζ). Eu T T An analogous argument to the one given in the proof of Proposition 3.1 shows that T T δz·ε = z δε for all z ∈ T and ε ∈ Eu. We claim that δT : γ ∈ G is a countable orthonormal set. Since G is a second- Su(γ) q(u) countable Hausdorff étalegroupoid, Gq(u) is discrete and countable. For all α, β ∈ Gq(u), we have ( 1 if α = β T T T δ δ = δ Su(β) = Su(α) Su(β) Su(α) 0 if α 6= β, and so δT : γ ∈ G is a countable orthonormal set. Su(γ) q(u) −1 Finally, let X be any subset of Gq(u) such that q (X) is measurable. To see that δT : γ ∈ X is maximal, suppose that f ∈ L2(X; q−1(X)) satisfies f δT = 0 for Su(γ) Su(γ) every γ ∈ X. We have Z f δT = f(η) δT (η) dλ (η) Su(γ) Su(γ) u q−1(X) Z Z = f(z · S (γ)) δT (z · S (γ)) dz = z z f(S (γ)) dz = f(S (γ)), u Su(γ) u u u T T −1 and so f(Su(γ)) = 0 for each γ ∈ X. Since q (X) = {z · Su(γ): z ∈ T, γ ∈ X} and since f is -equivariant, we deduce that f = 0. Thus δT : γ ∈ X is an orthonormal basis T Su(γ) 2 −1 for L (X; q (X)). Lemma 4.2. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0), 2 and let πu be the regular representation of Σc(G; E) on L (Gq(u); Eu). For each ε ∈ Eu, let T 2 (0) δε ∈ L (Gq(u); Eu) be defined as in Proposition 4.1. For each v ∈ E , let Sv : Gq(v) → Ev be a (not necessarily continuous) section for q|Ev . Then for each f ∈ Σc(G; E) and ε ∈ Eu, X π (f) δT = f(S (α)) δT . u ε r(ε) Sr(ε)(α) ε α∈Gq(r(ε))

Proof. Fix f ∈ Σc(G; E) and ε, ζ ∈ Eu. Then Z Z T −1 T −1 T πu(f) δε (ζ) = f(ζη ) δε (η) dλu(η) = f z · (ζε ) δε (z · ε) dz Eu T Z = z z f(ζε−1) dz = f(ζε−1). T T Given α ∈ Gq(r(ε)), we have δ (ζ) 6= 0 if and only if ζ = w · Sr(ε)(α) ε for some Sr(ε)(α) ε −1 (necessarily unique) w ∈ T, and in this case, Sr(ε)(α) = w · (ζε ). Therefore, X f(S (α)) δT (ζ) = fw · (ζε−1) δT w · S (α) ε r(ε) Sr(ε)(α) ε Sr(ε)(α) ε r(ε) α∈G q(r(ε)) −1 −1 T = w f(ζε ) w = f(ζε ) = πu(f) δε (ζ). Theorem 4.3. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0), G E and define F := f ∈ Σ (I ; I ): f| E ≡ 0 . u c Iu r r (a) Let Ju denote the closure of Fu in the reduced norm. Then Ju is an ideal of ∗ G E Cr (I ; I ), and there is a surjective homomorphism r ∗ G E r ∗ G E θu : Cr (I ; I )/Ju → Cr (Iq(u); Iu ) r r G E satisfying θ (f + J ) = f| E for all f ∈ Σ (I ; I ). u u Iu c A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 15

∗ G E (b) Let Ju denote the closure of Fu in the full norm. Then Ju is an ideal of C (I ; I ), and there is an isomorphism ∗ G E ∗ G E θu : C (I ; I )/Ju → C (Iq(u); Iu ) G E satisfying θ (f + J ) = f| E for all f ∈ Σ (I ; I ). u u Iu c r ∗ G E r ∗ G E Remark 4.4. Somewhat surprisingly, the map θu : Cr (I ; I )/Ju → Cr (Iq(u); Iu ) of The- orem 4.3(a) is not in general an isomorphism, unlike the analogue for the full C*-algebras in Theorem 4.3(b). In Theorem 4.10, we prove this by using an example introduced by Willett in [38] of a nonamenable groupoid whose full and reduced C*-algebras coincide. In order to prove Theorem 4.3, we need several preliminary results. We begin by r ∗ G E ∗ G E showing that Ju and Ju are ideals of Cr (I ; I ) and C (I ; I ), respectively. Lemma 4.5. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0), and G E G E define F := f ∈ Σ (I ; I ): f| E ≡ 0 . Then F is an algebraic ideal of Σ (I ; I ). u c Iu u c r Let Ju and Ju denote the closures of Fu in the reduced and full norms, respectively. Then r ∗ G E ∗ G E Ju is an ideal of Cr (I ; I ), and Ju is an ideal of C (I ; I ). G E Proof. It is clear that Fu is a linear subspace of Σc(I ; I ). To see that Fu is an algebraic G E E −1 E ideal, fix f ∈ Fu and g ∈ Σc(I ; I ). For all ε ∈ Iu , we have ε ∈ Iu , and so ∗ −1 ∗ E f (ε) = f(ε ) = 0. Thus f ∈ Fu. For all ε ∈ Iu , we have Z Z −1 u −1 (fg)(ε) = f(ζ) g(ζ ε) dλ (ζ) = 0 and (gf)(ε) = g(εζ ) f(ζ) dλu(ζ) = 0. E E Iu Iu G E Hence fg, gf ∈ Fu, and thus Fu is an algebraic ideal of Σc(I ; I ). Since all C*- r ∗ G E algebraic operations are continuous, it follows that Ju and Ju are ideals of Cr (I ; I ) ∗ G E and C (I ; I ), respectively. Lemma 4.6. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0). For E T G E E each ε ∈ Iu , define δε ∈ Σc Iq(u); Iu as in Proposition 3.1. Then for each ε ∈ Iu , there G E exists j ∈ Σ (I ; I ) such that j | E = δT and j = w j for each w ∈ . ε c ε Iu ε w·ε ε T Proof. By Corollary 2.11(b), IE is a twist over IG, and so by Lemma 2.5, we can find E G a local trivialisation (Bα,Pα, φPα )α∈IG for I such that for each α ∈ I , Bα is an open G E G bisection of I containing α. Fix ε ∈ Iu . Use Urysohn’s lemma to choose hq(ε) ∈ Cc(I ) such that supp(hq(ε)) ⊆ Bq(ε) and hq(ε)(q(ε)) = 1. Recall from Lemma 2.4(c) that for each −1 ζ ∈ q (Bq(ε)), there is a unique zζ ∈ T such that ζ = φPq(ε) (q(ζ), zζ ), and the map ζ 7→ zζ −1 −1 is continuous on q (Bq(ε)). Thus ζ 7→ zζ hq(ε)(q(ζ)) is a continuous map from q (Bq(ε)) E to C. Define jε : I → C by ( z z h (q(ζ)) if ζ ∈ q−1(B ) : ε ζ q(ε) q(ε) jε(ζ) = −1 0 if ζ∈ / q (Bq(ε)).

−1 Then supp(jε) ⊆ q supp(hq(ε)) = T · Pq(ε) supp(hq(ε)) . Since T and Pq(ε) supp(hq(ε)) E are compact sets and the action of T on I is continuous, supp(jε) is compact. Since −1 E −1 jε|q (Bq(ε)) is continuous and q (Bq(ε)) is open, jε is continuous on all of I . To see that jε E −1 −1 is T-equivariant, ﬁx ζ ∈ I and w ∈ T. Then w·ζ ∈ q (Bq(ε)) if and only if ζ ∈ q (Bq(ε)). −1 If ζ ∈ q (Bq(ε)), then zw·ζ = w zζ , and so jε(w · ζ) = zε w zζ hq(ε)(q(ζ)) = w jε(ζ). On the −1 G E other hand, if ζ∈ / q (Bq(ε)), then jε(w · ζ) = 0 = w jε(ζ). Hence jε ∈ Σc(I ; I ). E −1 We now show that j | E = δT. Fix ζ ∈ I . Suppose ﬁrst that ζ∈ / q (B ). Then ε Iu ε u q(ε) T q(ζ) 6= q(ε), and so ζ 6= w · ε for all w ∈ T. Hence jε(ζ) = 0 = δε (ζ). Now suppose that 16 BECKY ARMSTRONG

−1 ζ ∈ q (Bq(ε)). Since r|Bq(ε) (q(ζ)) = q(u) = r|Bq(ε) (q(ε)) and Bq(ε) is a bisection, we have q(ζ) = q(ε). Hence hq(ε)(q(ζ)) = 1, and ζ = zζ · Pq(ε)(q(ε)) = (zε zζ ) · ε. Therefore, T jε(ζ) = zε zζ hq(ε)(q(ζ)) = zε zζ = δε (ζ), and so j | E = δT. Finally, for all w ∈ , we have q(w · ε) = q(ε) and z = w z , and ε Iu ε T w·ε ε thus jw·ε = w jε.

We will use the following lemma to show that the map θu of Theorem 4.3(b) is injective. Lemma 4.7. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0), ∗ G E E and let Ju be the ideal of C (I ; I ) defined in Lemma 4.5. For each ε ∈ Iu , define T G E G E δε ∈ Σc Iq(u); Iu as in Proposition 3.1, and use Lemma 4.6 to choose jε ∈ Σc(I ; I ) E G E such that j | E = δT. For any ε ∈ I and k ∈ Σ (I ; I ) satisfying k | E = δT, we have ε Iu ε u ε c ε Iu ε ∗ G E jε − kε ∈ Ju. The quotient C*-algebra C (I ; I )/Ju is unital with identity ju + Ju, and ∗ G E each jε + Ju is a unitary element of C (I ; I )/Ju. Moreover, jεjζ + Ju = jεζ + Ju for all E ε, ζ ∈ Iu . E G E Proof. For any ε ∈ I and k ∈ Σ (I ; I ) satisfying k | E = δT, we have (j − k )| E ≡ 0, u ε c ε Iu ε ε ε Iu and hence jε − kε ∈ Ju. ∗ G E G E E We now show that C (I ; I )/Ju is unital. Fix f ∈ Σc(I ; I ). For all ξ ∈ Iu , we have Z Z Z −1 u T (juf)(ξ) = ju(η) f(η ξ) dλ (η) = δu (w·u) f w·(uξ) dw = w w f(ξ) dw = f(ξ). E Iu T T

Hence (j f −f)| E ≡ 0, and so j f −f ∈ J . A similar argument shows that fj −f ∈ J . u Iu u u u u Thus

(f + Ju)(ju + Ju) = fju + Ju = f + Ju = juf + Ju = (ju + Ju)(f + Ju), ∗ G E and so ju + Ju is the identity element of C (I ; I )/Ju. E ∗ G E Fix ε ∈ Iu . We will show that jε + Ju is a unitary element of C (I ; I )/Ju. For all E ξ ∈ Iu , we have Z Z ∗ −1 u T T −1 (jεjε )(ξ) = jε(η) jε(ξ η) dλ (η) = δε (w · ε) δε w · (ξ ε) dw E Iu T Z T −1 T −1 = w w δε (ξ ε) dw = δu (ξ ) = ju(ξ). T ∗ ∗ Hence (j j − j )| E ≡ 0, and so j j − j ∈ J . Therefore, ε ε u Iu ε ε u u ∗ ∗ (jε + Ju)(jε + Ju) = jεjε + Ju = ju + Ju, ∗ and a similar argument shows that (jε + Ju) (jε + Ju) = ju + Ju. Thus jε + Ju is a unitary. E E Finally, ﬁx ε, ζ ∈ Iu . For all ξ ∈ Iu , we have Z Z −1 u T T −1 (jεjζ )(ξ) = jε(η) jζ (η ξ) dλ (η) = δε (w · ε) δζ w · (ε ξ) dw E Iu T Z T −1 T −1 = w w δζ (ε ξ) dw = δζ (ε ξ) = jεζ (ξ), T and hence (j j − j )| E ≡ 0. Thus j j − j ∈ J , and so j j + J = j + J . ε ζ εζ Iu ε ζ εζ u ε ζ u εζ u G E Proof of Theorem 4.3. Fix f ∈ Σ (I ; I ). Then f| E is continuous. We claim that c Iu G E E E −1 f| E ∈ Σ I ; I . We have osupp(f| E ) = osupp(f) I , and so since I = r| (u) is Iu c q(u) u Iu ∩ u u IE E E E closed in I , supp(f) I is a compact subset of I . Hence supp(f| E ) ⊆ supp(f), and ∩ u u Iu E E E E so f| E ∈ C (I ). For all ε ∈ I and z ∈ , we have z · ε ∈ I if and only if ε ∈ I , and Iu c u T u u E G E so f| E (z · ε) = zf| E (ε) for all ε ∈ I and z ∈ . Hence f| E ∈ Σ I ; I . Iu Iu u T Iu c q(u) u A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 17

r G E ∗ G E r For part (a), define ψ :Σ (I ; I ) → C (I ; I ) by ψ (f) := f| E . Routine calcula- u c r q(u) u u Iu r r tions show that ψu is a ∗-homomorphism that vanishes on Fu. We now show that ψu is G E (0) bounded in the reduced norm. Fix f ∈ Σc(I ; I ). For each v ∈ E , let E I G E 2 G E Iv G E 2 G E πv :Σc(I ; I ) → B L Iq(v); Iv and ρ :Σc Iq(v); Iv → B L Iq(v); Iv G E G E be the regular representations associated to v of Σc(I ; I ) and Σc Iq(v); Iv , respectively, E (0) onto the space of square-integrable T-equivariant functions on Iv . For each v ∈ E , we I IE have π (f) = ρ v f| E , and hence v Iv r IE I (0) kψ (f)k = kf| E k = ρ v f| E ≤ sup π (f) : v ∈ E = kfk . u r Iu r Iu v r r G E ∗ G E r So ψu is bounded, and since Σc(I ; I ) is dense in Cr (I ; I ), ψu extends to a C*- r ∗ G E ∗ G E r homomorphism ψu : Cr (I ; I ) → Cr (Iq(u); Iu ). Recall from Lemma 4.5 that Ju is an ∗ G E r ideal of Cr (I ; I ). Since ψu is bounded and vanishes on Fu by definition, it vanishes on r r Ju. Therefore, ψu descends to a homomorphism r ∗ G E r ∗ G E θu : Cr (I ; I )/Ju → Cr (Iq(u); Iu ) r r G E r satisfying θ (f + J ) = f| E for all f ∈ Σ (I ; I ). We claim that θ is surjective. Since u u Iu c u G E ∗ G E G E Σc Iq(u); Iu is dense in Cr (Iq(u); Iu ), it suffices to show that Σc Iq(u); Iu is contained r in the image of θu. Recall from Proposition 3.1 that G E T E Σc Iq(u); Iu = span δε : ε ∈ Iu . E G E For each ε ∈ I , use Lemma 4.6 to choose j ∈ Σ (I ; I ) such that j | E = δT. Then for u ε c ε Iu ε E r r r each ε ∈ I , we have θ (j + J ) = j | E = δT, and hence θ is surjective. u u ε u ε Iu ε u G E ∗ G E For part (b), define ψ :Σ (I ; I ) → C (I ; I ) by ψ (f) := f| E . The argu- u c q(u) u u Iu ment used in part (a) shows that ψu is a ∗-homomorphism that vanishes on Fu. Thus ∗ G E Lemma 2.13 implies that ψu extends uniquely to a C*-homomorphism ψu : C (I ; I ) → ∗ G E ∗ G E C (Iq(u); Iu ). By Lemma 4.5, Ju is an ideal of C (I ; I ), and so a similar argument to the one used in part (a) shows that there is a surjective homomorphism ∗ G E ∗ G E θu : C (I ; I )/Ju → C (Iq(u); Iu ) G E satisfying θ (f + J ) = f| E for all f ∈ Σ (I ; I ). To see that θ is injective, recall u u Iu c u E G E from Lemma 4.6 that for each ε ∈ I , there exists j ∈ Σ (I ; I ) such that j | E = δT u ε c ε Iu ε E and jz·ε = z jε for each z ∈ T. By Lemma 4.7, jε + Ju : ε ∈ Iu is a family of unitary ∗ G E elements of C (I ; I )/Ju such that (jε +Ju)(jζ +Ju) = jεζ +Ju and jz·ε +Ju = z(jε +Ju) E for all ε, ζ ∈ Iu and z ∈ T. Hence Theorem 3.2 implies that there is a homomorphism ∗ G E ∗ G E ηu : C (Iq(u); Iu ) → C (I ; I )/Ju T E T G E satisfying ηu δε = jε + Ju for each ε ∈ Iu , where δε ∈ Σc Iq(u); Iu is defined as in ∗ G E Proposition 3.1. We claim that ηu ◦ θu is the identity map on C (I ; I )/Ju. To see G E ∗ G E this, observe that since f + Ju : f ∈ Σc(I ; I ) is a dense subspace of C (I ; I )/Ju and since ηu ◦ θu is continuous, it suffices to show that ηu θu(f + Ju) = f + Ju for all G E G E f ∈ Σc(I ; I ). Fix f ∈ Σc(I ; I ). Since G E T E θ (f + J ) = f| E ∈ Σ I ; I = span δ : ε ∈ I , u u Iu c q(u) u ε u E there is a finite subset F of I and a collection {c ∈ \{0} : ε ∈ F } such that f| E = u ε C Iu P T ε∈F cε δε . Since ηu is linear, we have X T X T X η θ (f + J ) = η f| E = η c δ = c η δ = c j + J . (4.1) u u u u Iu u ε ε ε u ε ε ε u ε∈F ε∈F ε∈F 18 BECKY ARMSTRONG

P P T P Since f − cε jε |IE = f|IE − cε δε ≡ 0, we have f − cε jε ∈ Ju. Thus ε∈F u u ε∈F ε∈F we deduce from equation (4.1) that ηu θu(f + Ju) = f + Ju. Therefore, ηu ◦ θu is the ∗ G E identity map on C (I ; I )/Ju, and so θu is injective. Corollary 4.8. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Fix u ∈ E (0). T ∗ G E Recall from Proposition 3.1 the definition of the identity element δu of both Cr (I ; I ) ∗ G E r ∗ G E and C (I ; I ). Recall from Theorem 4.3 the definitions of the ideals Ju of Cr (I ; I ) ∗ G E and Ju of C (I ; I ), and the maps r ∗ G E r ∗ G E ∗ G E ∗ G E θu : Cr (I ; I )/Ju → Cr (Iq(u); Iu ) and θu : C (I ; I )/Ju → C (Iq(u); Iu ).

r ∗ G E ∗ G E r r r (a) The map Qu : Cr (I ; I ) → Cr (Iq(u); Iu ) given by Qu(a) := θu(a + Ju) is a sur- G E (0) jective homomorphism. For every g ∈ Σc(I ; I ) satisfying q(supp(g)) ⊆ G and r T g(u) = 1, we have Qu(g) = δu . ∗ G E ∗ G E (b) The map Qu : C (I ; I ) → C (Iq(u); Iu ) given by Qu(a) := θu(a + Ju) is a sur- G E (0) jective homomorphism. For every g ∈ Σc(I ; I ) satisfying q(supp(g)) ⊆ G and T g(u) = 1, we have Qu(g) = δu . r Proof. We will only prove part (a), as the proof of part (b) is identical. The map Qu is a r surjective homomorphism because it is the composition of θu and the quotient map from ∗ G E ∗ G E r G E (0) Cr (I ; I ) to Cr (I ; I )/Ju. Suppose that g ∈ Σc(I ; I ) satisﬁes q(supp(g)) ⊆ G and r r r r r g(u) = 1. By the deﬁnition of Q and θ , we have Q (g) = θ (g + J ) = g| E . So we u u u u u Iu E −1 (0) (0) G must show that g| E = δT. Fix ε ∈ I . If ε ∈ q (G ), then q(ε) ∈ G ∩ I , and Iu u u q(u) so q(ε) = q(u). Thus ε = zε · u = i(q(u), zε) for some unique zε ∈ T, and so since g is T −1 (0) T-equivariant, we have g(ε) = g(zε · u) = zε g(u) = zε = δu (ε). If ε∈ / q (G ), then ε 6= z · u for all z ∈ , and so g(ε) = 0 = δT(ε). Therefore, g| E = δT, as required. T u Iu u

∗ G E In our proof of Theorem 4.3(b), we used the universal property of C (Iq(u); Iu ) given ∗ G E ∗ G E in Theorem 3.2 to show that θu : C (I ; I )/Ju → C (Iq(u); Iu ) is injective, but this argument doesn’t work in the reduced setting because the universal property doesn’t hold. In fact, somewhat surprisingly, even though θu is always an isomorphism, there r ∗ G E r ∗ G E exist examples of groupoids for which the map θu : Cr (I ; I )/Ju → Cr (Iq(u); Iu ) of Theorem 4.3(a) is not an isomorphism. One such example, due to Willett [38], comes from the class of HLS groupoids constructed in [21, Section 2] (see also [38, Deﬁnition 2.2]). Before presenting Willett’s example, we ﬁrst recall the construction of an HLS groupoid.

Suppose that (Kn)n∈N is an approximating sequence for a discrete group Γ, as defined in [38, Definition 2.1]. For each n ∈ N, define Γn := Γ/Kn, and Γ∞ := Γ. For each n ∈ N ∪ {∞}, denote the identity of the group Γn by eΓn . Define G G = {n} × Γn, n∈N∪{∞} and equip G with the groupoid operations coming from the group structure on the fibres (0) over each n ∈ N ∪ {∞}. Then G = {(n, g) ∈ G : g = eΓn }, and for all (n, g) ∈ G, we have r(n, g) = (n, eΓn ) = s(n, g). Thus Iso(G) = G. In [38, Definition 2.2], Willett endows this groupoid G with a topology, under which it is a second-countable locally compact Hausdorff étalegroupoid, called the HLS groupoid associated to the approximated group

(Γ, (Kn)n∈N).

Example 4.9. Let F2 denote the free group on two generators. For each n ∈ N, deﬁne \ Kn := {ker(φ): φ: F2 → Γ is a group homomorphism, |Γ| ≤ n}. (4.2) A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 19

By [38, Lemma 2.8], (Kn)n∈N is an approximating sequence for F2. Since F2 is not amenable, [38, Lemma 2.4] shows that the HLS groupoid G associated to the approximated ∗ group (F2, (Kn)n∈N) is not amenable. However, by [38, Lemmas 2.7 and 2.8], C (G) is ∗ isomorphic to Cr (G). Taking E to be the trivial twist G × T over the HLS groupoid G described in Exam- (0) r ∗ G E r ple 4.9, and u := (∞, eF2 ), 1 ∈ E , we now prove that the map θu : Cr (I ; I )/Ju → ∗ G E Cr (Iq(u); Iu ) of Theorem 4.3(a) is not injective. Note that since E is trivial, we omit it from G E G ∗ G E our notation in Theorem 4.10, and we identify Σc(I ; I ) with Cc(I ) and Cr (I ; I ) ∗ G with Cr (I ) via the ∗-isomorphism deﬁned in Proposition 2.14.

Theorem 4.10. For each n ∈ N, let Kn be deﬁned as in equation (4.2), and let G be the HLS groupoid associated to (F2, (Kn)n∈N), as described in Example 4.9. Consider the unit (0) r ∗ G r u := (∞, eF2 ) ∈ G , where eF2 is the identity element of F2. The map θu : Cr (I )/Ju → ∗ G Cr (Iu ) of Theorem 4.3(a) is not injective. G u ∼ Proof. Deﬁne Γ := F2. Since Iso(G) = G, we have Iu = Gu = {∞} × Γ∞ = Γ. Let ∗ G ∗ G Υ: C (I ) → Cr (I ) be the unique homomorphism that restricts to the identity map on G Cc(I ). By [38, Lemmas 2.7 and 2.8] and [2, Corollary 3.3.23], Υ is an isomorphism, and so G r the full and reduced C*-norms coincide on Cc(I ). Thus Υ|Ju : Ju → Ju is an isomorphism. ∗ G r Observe that for all a, b ∈ C (I ) with a − b ∈ Ju, we have Υ(a) − Υ(b) = Υ(a − b) ∈ Ju. ∗ G ∗ G r r Hence there is a map Υu : C (I )/Ju → Cr (I )/Ju satisfying Υu(a + Ju) = Υ(a) + Ju, ∗ G ∗ G ∗ G r for all a ∈ C (I ). Since Υ: C (I ) → Cr (I ) and Υ|Ju : Ju → Ju are isomorphisms, it is clear that Υu is also an isomorphism. ∗ G ∗ G Let Ξu : C (Iu ) → Cr (Iu ) be the unique homomorphism that restricts to the identity G map on Cc(Iu ). Since Γ = F2 is a nonamenable group, we know that Ξu is not injective. G For all f ∈ Cc(Iu ), we have r r r r r θ (Υ (f + J )) = θ (Υ(f) + J ) = θ (f + J ) = f| G = Ξ (f| G ) = Ξ (θ (f + J )). u u u u u u u Iu u Iu u u u r G Therefore, the maps θu ◦ Υu and Ξu ◦ θu agree on the set {f + Ju : f ∈ Cc(I )}, which is a ∗ G dense subspace of C (I )/Ju. Since these maps are homomorphisms of C*-algebras, they ∗ G r −1 are continuous, and hence they agree on all of C (I )/Ju. Therefore, θu = Ξu ◦ θu ◦ Υu . −1 r Since θu ◦ Υu is surjective and Ξu is not injective, it follows that θu is not injective.

5. Unique state extensions In this section we provide a sufficient compressibility condition under which a state (or pure state) of a C*-subalgebra B of a (not necessarily unital) C*-algebra A has a unique state (or pure state) extension to A. This result is an extension of a well known result of Anderson about unital C*-algebras (proved in the paragraph preceding [1, Theorem 3.2]), but we were unable to find a proof of it in the literature, and so we present our own. We begin by introducing some notation. Notation 5.1. Given a C*-algebra A and a state φ of A, we define ∗ Lφ := {a ∈ A : φ(a a) = 0},

Mφ := {a ∈ A : φ(ax) = φ(xa) = φ(a)φ(x) for all x ∈ A}, and Uφ := {a ∈ A : |φ(a)| = kak = 1}. For any subalgebra B of A, we write B∗ := {b∗ : b ∈ B}. Our compressibility condition is inspired by [1]. 20 BECKY ARMSTRONG

Deﬁnition 5.2. Let A be a C*-algebra and let B be a C*-subalgebra of A. Suppose that φ is a state of B. We say that A is B-compressible modulo φ if, for each a ∈ A and > 0, there exists b ∈ Uφ ⊆ B and c ∈ B such that b ≥ 0 and kbab − ck < . Theorem 5.3. Let A be a C*-algebra and let B be a C*-subalgebra of A. If φ is a state of B such that A is B-compressible modulo φ, then φ has a unique state extension to A. If φ is a pure state, then so is its unique extension. Before proving this theorem, we need the following two preliminary results.

Lemma 5.4. Let A be a unital C*-algebra with identity 1A, and let φ be a state of A. The φ-multiplicative set Mφ is a unital C*-subalgebra of A. Moreover, a ∈ Mφ if and ∗ only if a − φ(a)1A ∈ Lφ ∩ Lφ. Proof. A routine argument proves the first statement. (For instance, see the proof of [2, Lemma 5.2.4].) For the second statement, fix a ∈ A and define b := a − φ(a)1A. Then ∗ φ(b) = 0. Suppose that a ∈ Mφ. We must show that b, b ∈ Lφ. Since Mφ is a C*- ∗ ∗ ∗ ∗ algebra, we have b, b ∈ Mφ. Hence φ(b b) = φ(bb ) = φ(b)φ(b ) = φ(b)φ(b) = 0, and ∗ ∗ so b, b ∈ Lφ. For the converse, suppose that b = a − φ(a)1A ∈ Lφ ∩ Lφ. Fix x ∈ A. Since φ(b) = 0, we have φ(b)φ(x) = 0. Since φ(b∗b) = 0 = φ(bb∗), the Cauchy–Schwarz inequality implies that |φ(bx)|2 ≤ φ(bb∗)φ(x∗x) = 0 and |φ(xb)|2 ≤ φ(xx∗)φ(b∗b) = 0.

Thus φ(bx) = φ(xb) = 0 = φ(b)φ(x), and so b ∈ Mφ. Since Mφ is a unital C*-algebra, we have a = b + φ(a)1A ∈ Mφ. Lemma 5.5. Let A be a (not necessarily unital) C*-algebra, and let φ be a state of A. Then Uφ ⊆ Mφ.

Proof. We first suppose that A is unital with identity 1A. Fix a ∈ Uφ, and define b := ∗ a − φ(a)1A ∈ A. By Lemma 5.4, it suffices to prove that b ∈ Lφ ∩ Lφ. Since φ is positive, ∗ ∗ 2 2 we have φ(b b) ≥ 0. Since a ∈ Uφ, we have ka ak = kak = |φ(a)| = 1, and hence φ(b∗b) = φ(a∗a) − φ(a∗)φ(a) − φ(a)φ(a) + φ(a)φ(a) = φ(a∗a) − |φ(a)|2 ≤ ka∗ak − 1 = 0. ∗ ∗ Thus φ(b b) = 0, and so b ∈ Lφ. A similar argument shows that b ∈ Lφ, and so Uφ ⊆ Mφ. Now suppose that A is non-unital. Let A+ denote the minimal unitisation of A, and let φ be the unique state extension of φ to A+. Then U ⊆ M . Fix a ∈ U . Then e φe φe φ

φe(a, 0) = |φ(a)| = 1 = kak = k(a, 0)k, and hence (a, 0) ∈ U ⊆ M . Fix x ∈ A. Using that (a, 0) ∈ M , we see that φe φe φe φ(ax) = φe(ax, 0) = φe (a, 0)(x, 0) = φe (x, 0)(a, 0) = φe(xa, 0) = φ(xa), and φ(ax) = φe(ax, 0) = φe (a, 0)(x, 0) = φe(a, 0) φe(x, 0) = φ(a)φ(x), and hence a ∈ Mφ. Proof of Theorem 5.3. We know by [9, II.6.3.1] that every state of B extends to a state of A. Let φ be a state of B such that A is B-compressible modulo φ. To see that φ has a unique state extension to A, suppose that φ1 and φ2 are state extensions of φ to A. We must show that φ1 = φ2. Fix a ∈ A. For each n ∈ N, choose bn ∈ Uφ and cn ∈ B such 1 that bn ≥ 0 and kbnabn − ck < n . Then 0 ≤ φ(bn) = |φ(bn)| = 1 for all n ∈ N. Since

φ1|B = φ = φ2|B, Lemma 5.5 implies that Uφ ⊆ Uφ1 ∩ Uφ2 ⊆ Mφ1 ∩ Mφ2 . Therefore, for each n ∈ N, we have bn ∈ Mφ1 ∩ Mφ2 , and hence

φi(bnabn) = φi(bn) φi(a) φi(bn) = φ(bn) φi(a) φ(bn) = φi(a) for i ∈ {1, 2}. (5.1) A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 21

2 Fix > 0, and choose n ∈ N with n < . Then equation (5.1) implies that for i ∈ {1, 2}, 1 |φ (a) − φ(c )| = |φ (b ab ) − φ (c )| = |φ (b ab − c )| ≤ kb ab − c k < . i n i n n i n i n n n n n n n Hence we have 1 1 |φ (a) − φ (a)| ≤ |φ (a) − φ(c )| + |φ(c ) − φ (a)| < + < , 1 2 1 n n 2 n n and thus φ1(a) = φ2(a). Finally, if φ is a pure state of B, then [9, II.6.3.2] implies that the unique state extension of φ to A is also a pure state.

6. A uniqueness theorem for reduced twisted groupoid C*-algebras

In this section we prove that there is an embedding ιr of the twisted C*-algebra ∗ G E Cr (I ; I ) associated to the interior of the isotropy of a Hausdorff étalegroupoid G ∗ into Cr (G; E) (see Proposition 6.1). We use this result to prove our uniqueness theorem ∗ (Theorem 6.4), which states that a C*-homomorphism Ψ of Cr (G; E) is injective if and only if Ψ ◦ ιr is injective. We then use our uniqueness theorem to prove Corollary 6.10, ∗ which states that if G is effective, then Cr (G; E) is simple if and only if G is minimal. With the exception of Corollary 6.10, the results in this section are extensions of the results in [2, Section 5.3] to the setting of C*-algebras of groupoid twists. The reduced case of the following result is a generalisation of [31, Proposition 1.9] to the twisted setting, but the ideas used in the proof were inspired by the proofs of [13, Theorem 3.1(a)] and [34, Lemma 3]. Proposition 6.1. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. There is a homomorphism ι: C∗(IG; IE ) → C∗(G; E) such that ( f(ε) if ε ∈ IE ι(f)(ε) = 0 if ε∈ / IE

G E G E for all f ∈ Σc(I ; I ) and ε ∈ E. We have ι Σc(I ; I ) ⊆ Σc(G; E), and ι descends to ∗ G E ∗ G an injective homomorphism ιr : Cr (I ; I ) → Cr (G; E). If I is amenable, then ι is also injective. Before proving Proposition 6.1, we need the following preliminary result. Lemma 6.2. Let (E, i, q) be a twist over a Hausdorﬀ ´etalegroupoid G. Fix u ∈ E (0). There is an equivalence relation ∼ on Eu given by E E ε ∼ ζ if and only if εIu = ζIu . E For each ε ∈ Eu, the set εIu is the equivalence class of ε under ∼. If q(ε) = q(ζ) for some ε, ζ ∈ Eu, then ε ∼ ζ.

Proof. It is routine to check that ∼ is an equivalence relation on Eu. Fix ε ∈ Eu, and let [ε] denote the equivalence class of ε under ∼. We claim that E E E E [ε] = εIu . For all ζ ∈ [ε], we have ζ = ζu ∈ ζIu = εIu , and so [ε] ⊆ εIu . For the reverse E E E E containment, ﬁx η ∈ εIu . Then η = εξ for some ξ ∈ Iu , and so ηIu ⊆ εIu . Also, for all E −1 −1 E E E κ ∈ Iu , we have εκ = εξξ κ = η(ξ κ) ∈ ηIu , and hence εIu ⊆ ηIu . Thus η ∼ ε, and E E so η ∈ [ε]. Hence εIu ⊆ [ε], and thus εIu is the equivalence class of ε under ∼. Finally, suppose that q(ε) = q(ζ) for some ε, ζ ∈ Eu. Then ε = z · ζ for some z ∈ T, E and so ε = z · (ζu) = ζ(z · u) ∈ ζIu = [ζ]. Therefore, ε ∼ ζ. 22 BECKY ARMSTRONG

G E Proof of Proposition 6.1. We first show that the given formula satisfies ι Σc(I ; I ) ⊆ G E E Σc(G; E). Fix f ∈ Σc(I ; I ). Since I is open in E, a standard argument (see, for E instance, [2, Lemma 5.3.2]) shows that ι(f) ∈ Cc(I ). For all ε ∈ E and z ∈ T, we have z · ε ∈ IE if and only if ε ∈ IE , and so the T-equivariance of ι(f) follows from the G E T-equivariance of f. Hence ι Σc(I ; I ) ⊆ Σc(G; E). Routine calculations show that G E ∗ G E ι|Σc(I ;I ) :Σc(I ; I ) → C (G; E) is a ∗-homomorphism, and hence Lemma 2.13 shows that it extends uniquely to a C*-homomorphism ι: C∗(IG; IE ) → C∗(G; E). ∗ G E ∗ To see that ι descends to an injective homomorphism ιr : Cr (I ; I ) → Cr (G; E), it G E (0) suffices to show that kι(f)kr = kfkr for all f ∈ Σc(I ; I ). Fix u ∈ E , and let 2 I G E 2 G E πu :Σc(G; E) → B L (Gq(u); Eu) and πu :Σc(I ; I ) → B L Iq(u); Iu G E be the regular representations associated to u of Σc(G; E) and Σc(I ; I ) onto the spaces E of square-integrable T-equivariant functions on Eu and Iu , respectively. Recall from Lemma 6.2 that there is an equivalence relation ∼ on Eu under which the equivalence class of ε ∈ E is the set εIE . Choose a transversal T for ∼ containing u. Then E = S εIE , u u u u ε∈Tu u E E and εIu ∩ ζIu = ∅ for all distinct ε, ζ ∈ Tu. Hence 2 M 2 G E B L (Gq(u); Eu) = B L q(ε)Iq(u); εIu . ε∈Tu 2 G E Fix ε ∈ Eu. We claim that L q(ε)Iq(u); εIu is invariant under bounded linear operators G E G E E −1 G in πu ι(Σc(I ; I )) . To see this, fix f ∈ Σc(I ; I ). Since εIu = q q(ε)Iq(u) , it is a closed, and in particular, measurable, subset of Eu. So Proposition 4.1 implies that the T E 2 G E collection δη : η ∈ εIu is an orthonormal basis for L q(ε)Iq(u); εIu , and hence it T 2 G E E E suffices to show that πu(ι(f)) δη ∈ L q(ε)Iq(u); εIu for each η ∈ εIu . Fix η ∈ εIu , and (0) for each v ∈ E , let Sv : Gq(v) → Ev be a (not necessarily continuous) section for q|Ev . Then (0) E for each v ∈ E , Sv| G is a section for q|IE . Since r(η) = r(ε) and osupp(ι(f)) ⊆ I , Iq(v) v Lemma 4.2 implies that X X π (ι(f)) δT = ι(f)(S (α)) δT = f(S (α)) δT . (6.1) u η r(η) Sr(η)(α) η r(ε) Sr(ε)(α) η G α∈Gq(r(η)) α∈Iq(r(ε))

T 2 G E E To see that πu(ι(f)) δη ∈ L q(ε)Iq(u); εIu , we ﬁrst show that εIu is invariant under left E E E multiplication by elements of Ir(ε). Since η ∈ εIu , there exists ξ ∈ Iu such that η = εξ. E G Fix λ ∈ Ir(ε). Since q(ε) ∈ Gu and q(λ) ∈ Iq(r(ε)),[37, Proposition 2.5(b)] implies that −1 G −1 E −1 E q(ε λε) ∈ Iq(u), and hence ε λε ∈ Iu . Thus λη = λ(εξ) = ε(ε λεξ) ∈ εIu . Now, for G E E each α ∈ Iq(r(ε)), we have Sr(ε)(α) ∈ Ir(ε). So by the above argument, Sr(ε)(α) η ∈ εIu , T 2 G E and hence equation (6.1) implies that πu(ι(f)) δη ∈ L q(ε)Iq(u); εIu , as claimed. Fix f ∈ Σ (IG; IE ). Then, since E = S εIE , we have c u ε∈Tu u M πu(ι(f)) = πu(ι(f)) 2 G E L (q(ε)Iq(u);εIu ) ε∈Tu n o

= sup πu(ι(f)) 2 G E : ε ∈ Tu L (q(ε)Iq(u);εIu ) n o

= sup πu(ι(f)) 2 G E : ε ∈ Eu . (6.2) L (q(ε)Iq(u);εIu ) E Fix ε ∈ Eu. We saw earlier that εIu is invariant under left multiplication by elements of E G Ir(ε). A similar argument shows that q(ε)Iq(u) is invariant under left multiplication by G G G elements of Iq(r(ε)). Thus ϕq(ε) : α 7→ αq(ε) is a bijection of Iq(r(ε)) onto q(ε)Iq(u), with A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 23

−1 −1 G inverse ϕq(ε) : β 7→ βq(ε) . Choose a (not necessarily continuous) section Sr(ε) : Iq(r(ε)) → E G E −1 I for q| E , and deﬁne Su : q(ε)I → εI by Su(γ) := Sr(ε)(ϕ (γ)) ε. Then Su r(ε) Ir(ε) q(u) u q(ε) G is a section for q| E . By Proposition 4.1, the collection δT : γ ∈ I is an εIu Sr(ε)(γ) q(r(ε)) orthonormal basis for L2IG ; IE , and the collection δT : γ ∈ q(ε)IG is an q(r(ε)) r(ε) Su(γ) q(u) 2 G E orthonormal basis for L q(ε)Iq(u); εIu . Thus, by [30, Proposition 3.1.14], there is a unitary operator 2 G E 2 G E Uε : L Iq(r(ε)); Ir(ε) → L q(ε)Iq(u); εIu satisfying U δT = δT = δT , for each γ ∈ IG . ε Sr(ε)(γ) Su(ϕq(ε)(γ)) Sr(ε)(γ) ε q(r(ε)) G In fact, Proposition 4.1 implies that for each z ∈ T and γ ∈ Iq(r(ε)), we have U δT = z U δT = z δT = δT , ε z·Sr(ε)(γ) ε Sr(ε)(γ) Sr(ε)(γ) ε (z·Sr(ε)(γ)) ε

T T ∗ T T E E and hence Uε δζ = δζε and Uε δη = δηε−1 , for each ζ ∈ Ir(ε) and η ∈ εIu . E Fix ζ ∈ Ir(ε). Using Lemma 4.2 for the second and ﬁnal equalities, we see that

∗ T ∗ T Uε πu(ι(f)) Uε δζ = Uε πu(ι(f)) δζε X = U ∗ ι(f)(S (α)) δT ε r(ζ) Sr(ζ)(α) ζε α∈Gq(r(ζ)) X = f(S (β)) U ∗ δT r(ζ) ε Sr(ζ)(β) ζε G β∈Iq(r(ζ)) X = f(S (β)) δT r(ζ) Sr(ζ)(β) ζ G β∈Iq(r(ζ)) I T = πr(ε)(f) δζ . ∗ I Therefore, Uε πu(ι(f)) Uε = πr(ε)(f), and so

∗ I πu(ι(f)) 2 G E = Uε πu(ι(f)) Uε = πr(ε)(f) . (6.3) L (q(ε)Iq(u);εIu ) Together, equations (6.2) and (6.3) imply that I πu(ι(f)) = sup πr(ε)(f) : ε ∈ Eu , and hence (0) kι(f)kr = sup πu(ι(f)) : u ∈ E I (0) = sup sup πr(ε)(f) : ε ∈ Eu : u ∈ E I (0) = sup πv (f) : v ∈ E

= kfkr. (6.4) For the final claim, suppose that IG is amenable. Define (0) D0 := f ∈ Σc(G; E): q(supp(f)) ⊆ G and I G E (0) D0 := f ∈ Σc(I ; I ): q(supp(f)) ⊆ G . I I Let D and D denote the completions of D0 and D0 , respectively, with respect to the full norm. We claim that ι|DI is injective. To see this, it suffices to show that kfk ≤ kι(f)k I I for all f ∈ D0 , because then ι|DI is isometric, and hence injective. Fix f ∈ D0 . By [35, 24 BECKY ARMSTRONG

Theorem 11.1.11], we have kfk = kfkr, and so by equation (6.4), we have kfk = kfkr = kι(f)kr ≤ kι(f)k, as required. Let ∗ I ∗ G E I Φ0 : C (G; E) → D and Φ0 : C (I ; I ) → D be the conditional expectations extending restriction of functions to q−1(G(0)). Since IG is G E I amenable, the full and reduced norms agree on Σc(I ; I ), and so Φ0 is faithful. A routine I calculation shows that Φ0 ◦ ι = ι ◦ Φ0 , and since ι|DI is injective, a standard argument ∗ G E (see, for instance, [36, Lemma 3.14]) shows that ι is injective on C (I ; I ). G We now prove that if I is closed, then the map that restricts functions in Σc(G; E) E ∗ ∗ G E to I extends to a conditional expectation from Cr (G; E) to ιr Cr (I ; I ) , and it also extends to a conditional expectation from C∗(G; E) to ιC∗(IG; IE ) if IG is amenable. Lemma 6.3. Let G be a Hausdorff étalegroupoid such that IG is closed in G, and let G E (E, i, q) be a twist over G. For all f ∈ Σc(G; E), we have f|IE ∈ Σc(I ; I ). ∗ G E ∗ (a) Let ιr : Cr (I ; I ) → Cr (G; E) be the homomorphism of Proposition 6.1. There I ∗ ∗ G E I is a conditional expectation Ψr : Cr (G; E) → ιr Cr (I ; I ) satisfying Ψr (f) = I ιr(f|IE ) for all f ∈ Σc(G; E), and Ψr ◦ ιr = ιr. (b) Let ι: C∗(IG; IE ) → C∗(G; E) be the homomorphism of Proposition 6.1. If IG is amenable, then there is a conditional expectation ΨI : C∗(G; E) → ιC∗(IG; IE ) I I satisfying Ψ (f) = ι(f|IE ) for all f ∈ Σc(G; E), and Ψ ◦ ι = ι. E Proof. For all f ∈ Σc(G; E), f|IE is continuous, and osupp(f|IE ) = osupp(f) ∩ I . Since E −1 G E I = q (I ) is closed, supp(f|IE ) is a closed subset of supp(f), and so f|IE ∈ Cc(I ). For E E all ε ∈ E and z ∈ T, we have z · ε ∈ I if and only if ε ∈ I , and so f|IE (z · ε) = zf|IE (ε) E G E for all ε ∈ I and z ∈ T. Hence f|IE ∈ Σc(I ; I ). ∗ G E I I For (a), let Mr := ιr Cr (I ; I ) , and define Ψr :Σc(G; E) → Mr by Ψr (f) := ιr(f|IE ). I I (0) It is clear that Ψr is linear. We claim that Ψr is bounded. To see this, fix u ∈ E , and let 2 I G E 2 G E πu :Σc(G; E) → B L (Gq(u); Eu) and πu :Σc(I ; I ) → B L Iq(u); Iu G E be the regular representations associated to u of Σc(G; E) and Σc(I ; I ) onto the spaces E of square-integrable T-equivariant functions on Eu and Iu , respectively. Fix f ∈ Σc(G; E). 2 2 G E Let P ∈ B L (Gq(u); Eu) be the orthogonal projection onto L Iq(u); Iu ; that is, for each 2 g ∈ L (Gq(u); Eu) and ε ∈ E, define ( g(ε) if ε ∈ IE : u P (g)(ε) = E 0 if ε∈ / Iu . Then 2 P πu(f) P = sup P πu(f) P g : g ∈ L (Gq(u); Eu), kgk ≤ 1 2 = sup f ∗ (g|IE ) |IE : g ∈ L (Gq(u); Eu), kgk ≤ 1 n 2 G E o = sup (f|IE ) ∗ h : h ∈ L Iq(u); Iu , khk ≤ 1 I = πu (f|IE ) . (6.5) Since P is a projection, we have kP k ≤ 1, and hence equation (6.5) implies that I (0) kf|IE kr = sup πu (f|IE ) : u ∈ E (0) = sup P πu(f) P : u ∈ E (0) ≤ sup πu(f) : u ∈ E A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 25

= kfkr. (6.6)

Since ιr is bounded, we deduce from equation (6.6) that I E E Ψr (f) r = kιr(f|I )kr ≤ kf|I kr ≤ kfkr. I Thus Ψr :Σc(G; E) → Mr is a bounded linear map, and so it extends to a bounded linear I ∗ I I map Ψr : Cr (G; E) → Mr satisfying Ψr = kΨr k. For convenience, we will henceforth I I G E E I refer to Ψr as Ψr . For all f ∈ ιr Σc(I ; I ) , we have supp(f) ⊆ I , and hence Ψr (f) = G E I ιr(f|IE ) = f. Thus, since ιr Σc(I ; I ) is dense in Mr, we have Ψr (f) = f for all I I f ∈ Mr, and so Ψr ◦ιr = ιr. Hence Ψr is a projection, and since Mr is nontrivial, we have I I kΨr k ≥ 1. Therefore, [9, Theorem II.6.10.2] implies that Ψr is a conditional expectation. ∗ G E I I For (b), let M := ι C (I ; I ) , and define Ψ :Σc(G; E) → M by Ψ (f) := ι(f|IE ). We will only show that ΨI is bounded, because the remainder of the proof is analogous G to the proof of part (a). The amenability of I implies that kf|IE k = kf|IE kr for each f ∈ Σc(G; E). Hence, using equation (6.6), and the fact that ι is bounded, we see that I Ψ (f) = kι(f|IE )k ≤ kf|IE k = kf|IE kr ≤ kfkr ≤ kfk, as required. We now present our main result: a uniqueness theorem for reduced twisted groupoid C*-algebras, which generalises [12, Theorem 3.1(b)]. Theorem 6.4 (C*-uniqueness theorem). Let (E, i, q) be a twist over a Hausdorff étale ∗ G E ∗ groupoid G. Let ιr : Cr (I ; I ) → Cr (G; E) be the injective homomorphism of Propo- ∗ G E sition 6.1, and define Mr := ιr Cr (I ; I ) . Suppose that A is a C*-algebra and that ∗ Ψ: Cr (G; E) → A is a C*-homomorphism. Then Ψ is injective if and only if Ψ ◦ ιr is an ∗ G E injective C*-homomorphism of Cr (I ; I ). In order to prove Theorem 6.4, we need the following preliminary result, which is an extension of [12, Theorem 3.1(a)] to the twisted setting. Proposition 6.5. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Let ∗ G E ∗ ∗ G E ∗ ιr : Cr (I ; I ) → Cr (G; E) and ι: C (I ; I ) → C (G; E) be the homomorphisms of Proposition 6.1, and define ∗ G E ∗ G E Mr := ιr Cr (I ; I ) and M := ι C (I ; I ) . (0) u E Suppose that u ∈ E satisfies Eu = Iu . ∗ q(u) u (a) If ϕr is a state of Mr such that ϕr ◦ ιr factors through Cr Gq(u) ; Eu , then ϕr has ∗ a unique state extension to Cr (G; E). ∗ q(u) u (b) If ϕ is a state of M such that ϕ ◦ ι factors through C Gq(u) ; Eu , then ϕ has a unique state extension to C∗(G; E). In order to prove Proposition 6.5, we need the following two preliminary results. The first of these results is a generalisation of [12, Lemma 3.3(b)] to the twisted setting. Lemma 6.6. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose that (0) u E G E u ∈ E satisfies Eu = Iu . For each f ∈ Σc(G; E), there exists g ∈ Σc(I ; I ) satisfying (0) E g ≥ 0, q(supp(g)) ⊆ G , kgk = kgkr = g(u) = 1, and supp(gfg) ⊆ I .

u E q(u) G Proof. First observe that since Eu = Iu , we have Gq(u) = Iq(u). Fix f ∈ Σc(G; E). By P Lemma 2.12, we can write f = D∈F fD, where F is a ﬁnite collection of open bisections of G such that for each D ∈ F , fD ∈ Σc(G; E) and q(supp(fD)) ⊆ D. Choose open 26 BECKY ARMSTRONG

(0) neighbourhoods {VD ⊆ G : D ∈ F } of q(u) as in the proof of [12, Lemma 3.3(b)], so G that (VD DVD) ∩ q(supp(fD)) ⊆ I for each D ∈ F . Let V := ∩D∈F VD. Then V is an open neighbourhood of q(u) contained in G(0). Now use Urysohn’s lemma to choose (0) b ∈ Cc(V ) such that b ≥ 0 and b(q(u)) = kbk∞ = 1. Since G is a bisection of G, it follows that kbk = kbkr = kbk∞ = 1 (see, for instance, [35, Corollary 9.3.4]). For each −1 (0) (0) ε ∈ q (G ), there are unique elements vε ∈ G and zε ∈ T such that ε = i(vε, zε). Define g : IE → C by ( z b(v ) if ε ∈ q−1(G(0)) g(ε) := ε ε 0 if ε∈ / q−1(G(0)). It follows immediately from our choice of b and construction of g that g(u) = b(q(u)) = 1 and that q(supp(g)) ⊆ V ⊆ G(0). By [35, Lemma 11.1.9], there is an isomorphism from (0) I G E (0) Cc(G ) to D0 := {h ∈ Σc(I ; I ): q(supp(h)) ⊆ G } that maps b to g, and thus G E g ∈ Σc(I ; I ) and g ≥ 0. By [35, Theorem 11.1.11], this isomorphism extends to (0) I (isometric) isomorphisms of C0(G ) onto the full and reduced C*-completions of D0 , and therefore, kgk = kgkr = kbk∞ = 1. Finally, for each D ∈ F , we have G q(supp(gfDg)) ⊆ q(supp(g)) q(supp(fD)) q(supp(g)) ⊆ (VDV ) ∩ q(supp(fD)) ⊆ I P −1 G E by construction, and since f = D∈F fD, it follows that supp(gfg) ⊆ q (I ) = I . The next result is an extension of [12, Lemma 3.5] to the twisted setting. Lemma 6.7. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Let ∗ G E ∗ ∗ G E ∗ ιr : Cr (I ; I ) → Cr (G; E) and ι: C (I ; I ) → C (G; E) be the homomorphisms of Proposition 6.1, and define ∗ G E ∗ G E Mr := ιr Cr (I ; I ) and M := ι C (I ; I ) . (0) u E Suppose that u ∈ E satisfies Eu = Iu . ∗ (a) Fix > 0 and a ∈ Cr (G; E). There exist b, c ∈ Mr satisfying b ≥ 0, kbkr = 1, kbab − ckr < , and ϕr(b) = 1 for every state ϕr of Mr such that ϕr ◦ ιr factors ∗ q(u) u through Cr Gq(u) ; Eu . If a is positive, then c can be taken to be positive. (b) Fix > 0 and a ∈ C∗(G; E). There exist b, c ∈ M satisfying b ≥ 0, kbk = 1, kbab − ck < , and ϕ(b) = 1 for every state ϕ of M such that ϕ ◦ ι factors through ∗ q(u) u C Gq(u) ; Eu . If a is positive, then c can be taken to be positive. Proof. Both parts follow from Corollary 4.8 in the same way, and so we will only prove u E q(u) G part (a). First observe that since Eu = Iu , we have Gq(u) = Iq(u). Let

r ∗ G E ∗ G E ∗ q(u) u Qu : Cr (I ; I ) → Cr (Iq(u); Iu ) = Cr Gq(u) ; Eu ∗ be the surjective homomorphism of Corollary 4.8(a). Since Σc(G; E) is dense in Cr (G; E), we can choose f ∈ Σc(G; E) such that ka−fkr < . If a is positive, a standard C*-algebraic argument (see, for instance, [2, Lemma 5.3.10]) shows that f can also be taken to be G E (0) positive. Use Lemma 6.6 to choose g ∈ Σc(I ; I ) satisfying g ≥ 0, q(supp(g)) ⊆ G , E kgk = kgkr = g(u) = 1, and supp(gfg) ⊆ I . Deﬁne b := ιr(g) and c := bfb. Then E supp(c) ⊆ I , b, c ∈ Mr, b ≥ 0, and b(u) = g(u) = 1. If f ≥ 0, then it follows that c ≥ 0. (0) Since q(supp(b)) ⊆ G ,[35, Theorem 11.1.11] implies that kbk = kbkr. Moreover, since Proposition 6.1 implies that ιr is isometric, we have kbk = kbkr = kιr(g)kr = kgkr = 1, and hence 2 kbab − ckr = kbab − bfbkr ≤ kbkr ka − fkr = ka − fkr < ε. A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 27

∗ q(u) u Suppose that ϕr is a state of Mr such that ϕr ◦ ιr factors through Cr Gq(u) ; Eu . Then ∗ q(u) u r there is a state ψr of Cr Gq(u) ; Eu such that ϕr ◦ ιr = ψr ◦ Qu. By Corollary 4.8(a), we r T ∗ q(u) u have Qu(g) = δu , which is the identity element of Cr Gq(u) ; Eu defined in Proposition 3.1. r T Thus, since ψr is unital, we have ϕr(b) = ϕr(ιr(g)) = ψr(Qu(g)) = ψr(δu ) = 1. Proof of Proposition 6.5. Both parts follow from Lemma 6.7 in the same way, and so we will only prove part (a). Suppose that ϕr is a state of Mr such that ϕr ◦ ιr factors ∗ q(u) u ∗ through Cr Gq(u) ; Eu . Recalling the terminology defined in Definition 5.2, Cr (G; E) is Mr-compressible modulo ϕr by Lemma 6.7(a), and so by Theorem 5.3, ϕr has a unique ∗ state extension to Cr (G; E). We need the following two additional results before we prove Theorem 6.4. Lemma 6.8. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose that (0) u E r ∗ G E ∗ G E ∗ q(u) u u ∈ E satisfies Eu = Iu , and let Qu : Cr (I ; I ) → Cr (Iq(u); Iu ) = Cr Gq(u) ; Eu be ∗ G E ∗ the surjective homomorphism of Corollary 4.8(a). Let ιr : Cr (I ; I ) → Cr (G; E) be the ∗ G E injective homomorphism of Proposition 6.1, and define Mr := ιr Cr (I ; I ) . Let φ be a ∗ q(u) u r state of Cr Gq(u) ; Eu , and define ψ : Mr → C by ψ(ιr(a)) := φ(Qu(a)). Then ψ is a state ∗ G E ∗ q(u) u of Mr, and ψ ◦ιr is a state of Cr (I ; I ) that factors through Cr Gq(u) ; Eu . If φ is a pure state, then ψ and ψ ◦ ιr are also pure states.

r Proof. Since Qu is a C*-homomorphism and φ is a state, it is clear that ψ and ψ ◦ ιr are positive bounded linear functionals. To see that ψ and ψ ◦ ιr are states, we must G E show that kψk = kψ ◦ ιrk = 1. Using Lemma 6.6, we can ﬁnd g ∈ Σc(I ; I ) satisfying (0) r q(supp(g)) ⊆ G and g(u) = 1, and hence Corollary 4.8(a) implies that Qu(g) is the ∗ q(u) u ∗ q(u) u identity element of Cr Gq(u) ; Eu . Therefore, since φ is a state of Cr Gq(u) ; Eu , we have r kψk ≥ kψ ◦ ιrk ≥ |ψ(ιr(g))| = |φ(Qu(g))| = 1, r and so kψk = kψ ◦ ιrk = 1. Thus ψ is a state of Mr, and ψ ◦ ιr = φ ◦ Qu is a state of ∗ G E ∗ q(u) u Cr (I ; I ) that factors through Cr Gq(u) ; Eu . Suppose now that φ is a pure state. We claim that ψ ◦ ιr is a pure state. To see this, ∗ G E suppose that ψ1 and ψ2 are states of Cr (I ; I ) such that ψ ◦ ιr = tψ1 + (1 − t)ψ2 for some t ∈ (0, 1). We must show that ψ ◦ ιr = ψ1 = ψ2. We ﬁrst claim that ψ1 and ψ2 ∗ q(u) u r factor through Cr Gq(u) ; Eu . To see this, we will show that ker(Qu) ⊆ ker(ψ1) ∩ ker(ψ2). r For all a ∈ ker(Qu) such that a ≥ 0, we have ψ1(a), ψ2(a) ≥ 0, and r 0 ≤ tψ1(a) + (1 − t)ψ2(a) = ψ(ιr(a)) = φ(Qu(a)) = φ(0) = 0, r and hence ψ1(a) = 0 = ψ2(a). Since the kernel of Qu is a C*-algebra, it is spanned by its r positive elements, and so we deduce that ker(Qu) ⊆ ker(ψ1) ∩ ker(ψ2). Hence there exist ∗ q(u) u r r states φ1 and φ2 of Cr Gq(u) ; Eu such that ψ1 = φ1 ◦ Qu and ψ2 = φ2 ◦ Qu. We have r φ ◦ Qu = ψ ◦ ιr = tψ1 + (1 − t)ψ2 r r r = t(φ1 ◦ Qu) + (1 − t)(φ2 ◦ Qu) = (tφ1 + (1 − t)φ2) ◦ Qu. (6.7) r Since Qu is surjective, we deduce from equation (6.7) that φ = tφ1 + (1 − t)φ2. Hence φ = φ1 = φ2, because φ is a pure state. Therefore, r r r r ψ1 = φ1 ◦ Qu = φ ◦ Qu = ψ ◦ ιr and ψ2 = φ2 ◦ Qu = φ ◦ Qu = ψ ◦ ιr, and so ψ ◦ ιr is a pure state. A similar argument shows that ψ is also a pure state. 28 BECKY ARMSTRONG

Before we present the next result, we recall from [33, Proposition 4.3] the existence of ∗ ∗ (0) −1 (0) ∼ (0) the faithful conditional expectations Φr : Cr (G; E) → Cr G ; q (G ) = C0(G ) and I ∗ G E ∗ (0) −1 (0) ∼ (0) Φr : Cr (I ; I ) → Cr G ; q (G ) = C0(G ) extending restriction of functions. Lemma 6.9. Let (E, i, q) be a twist over a Hausdorff étalegroupoid G. Suppose that (0) u E ∗ G E ∗ u ∈ E satisfies Eu = Iu . Let ιr : Cr (I ; I ) → Cr (G; E) be the injective homomorphism ∗ G E of Proposition 6.1, and define Mr := ιr Cr (I ; I ) . Let evu be the evaluation map ∗ (0) −1 (0) ∼ (0) f 7→ f(u) on Cr G ; q (G ) = C0(G ). Then I ∗ G E ∗ q(u) u (a) evu ◦ Φr is a state of Cr (I ; I ) that factors through Cr Gq(u) ; Eu ; I (b) evu ◦ (Φr|Mr ) is a state of Mr that satisfies evu ◦ (Φr|Mr ) ◦ ιr = evu ◦ Φr ; and ∗ (c) evu ◦ Φr is the unique state extension of evu ◦ (Φr|Mr ) to Cr (G; E). I Proof. It is clear that evu ◦ Φr , evu ◦ (Φr|Mr ), and evu ◦ Φr are positive bounded linear functionals since they are composed of positive bounded linear maps. To see that they G E are states, use Lemma 6.6 to find g ∈ Σc(I ; I ) such that g(u) = 1. Then I Φr|Mr (ιr(g))(u) = Φr (g)(u) = |g(u)| = 1, and it follows that I evu ◦ Φr = kevu ◦ (Φr|Mr )k = kevu ◦ Φrk = 1. I Thus evu ◦ Φr , evu ◦ (Φr|Mr ), and evu ◦ Φr are states. I G E ∗ G E Since evu ◦ (Φr|Mr ) ◦ ιr and evu ◦ Φr agree on Σc(I ; I ), which is dense in Cr (I ; I ), I it follows that evu ◦ (Φr|Mr ) ◦ ιr = evu ◦ Φr . Thus part (b) holds. G −1 E r ∗ G E For part (a), define H := Iq(u) and EH := q (H) = Iu . Let Qu : Cr (I ; I ) → ∗ ∗ q(u) u Cr (H; EH) = Cr Gq(u) ; Eu be the surjective homomorphism of Corollary 4.8(a), and H ∗ ∗ (0) −1 (0) ∼ (0) let Φr : Cr H; EH → Cr H ; q (H ) = C0(H ) be the conditional expectation I ∗ extending restriction of functions. To see that evu ◦ Φr factors through Cr (H; EH), we ∗ r I (0) will find a state φu of Cr (H; EH) such that φu ◦ Qu = evu ◦ Φr . We have H = {q(u)}, H ∗ and by a similar argument to the one above, φu := evu ◦ Φr is a state of Cr H; EH . G E r I For all f ∈ Σ (I ; I ), we have (φ ◦ Q )(f) = φ f| E = f(u) = (ev ◦ Φ )(f). Since c u u u Iu u r G E ∗ G E r I Σc(I ; I ) is dense in Cr (I ; I ), it follows that φu ◦ Qu = evu ◦ Φr , as required. I We conclude by proving part (c). By parts (a) and (b), evu ◦ (Φr|Mr ) ◦ ιr = evu ◦ Φr ∗ G E ∗ q(u) u is a state of Cr (I ; I ) that factors through Cr Gq(u) ; Eu . Therefore, Proposition 6.5(a) ∗ implies that evu ◦ (Φr|Mr ) extends uniquely to a state of Cr (G; E). Thus, since evu ◦ Φr is ∗ an extension of evu ◦ (Φr|Mr ) to Cr (G; E), it must be the unique state extension.

Proof of Theorem 6.4. Since ιr is injective, it is clear that if Ψ is injective, then so is the homomorphism Ψ ◦ ιr. We prove the converse. For this, suppose that Ψ ◦ ιr is injective. ∗ Then Ψ is injective on the subalgebra Mr of Cr (G; E). Let E (0) u E G E (0) x G X := {u ∈ E : Eu = Iu } and X := q(X ) = {x ∈ G : Gx = Ix }. E For each u ∈ X , let Su be the collection of pure states ϕ of Mr such that ϕ ◦ ιr is ∗ G E ∗ q(u) u a pure state of Cr (I ; I ) that factors through Cr Gq(u) ; Eu . Define S := ∪u∈X E Su. ∗ By Proposition 6.5(a), each ϕ ∈ S extends uniquely to a state ϕ of Cr (G; E). For each ∗ ϕ ∈ S, let πϕ be the GNS representation of Cr (G; E) associated to ϕ. To see that Ψ is ∗ L injective on Cr (G; E), it suffices by [12, Theorem 3.2], to show that πS := ϕ∈S πϕ is ∗ ∗ faithful on Cr (G; E). For this, fix a ∈ Cr (G; E) such that πS (a) = 0. Then πϕ(a) = 0 ∗ ∗ (0) −1 (0) for every ϕ ∈ S. Let Φr : Cr (G; E) → Cr G ; q (G ) be the faithful conditional expectation extending restriction of functions. To see that a = 0, it suffices to show that ∗ ∗ Φr(a a) = 0, because Φr is faithful. Suppose, for contradiction, that Φr(a a) 6= 0. Then A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 29

∗ (0) ∗ Φr(a a) > 0, because Φr is positive. Let Ξa ∈ C0(G ) be the image of Φr(a a) under the ∗ (0) −1 (0) (0) isomorphism from Cr G ; q (G ) to C0(G ) given in [35, Theorem 11.1.11]. Then ∗ −1 Ξa > 0, since Φr(a a) > 0. Let Ya := Ξa (0, ∞) . Since Ξa is continuous, Ya is an open subset of G(0). Thus, since X G is dense in G(0) by [12, Lemma 3.3(a)], we have G G −1 G E Ya ∩ X 6= ∅. Choose x ∈ Ya ∩ X , and let u := i(x, 1). Then u ∈ (q|E(0) ) (X ) = X , ∗ and Φr(a a)(u) = Ξa(x) > 0. Fix > 0 such that ∗ Φr(a a)(u) > . (6.8) u E ∗ Since Eu = Iu and a a ≥ 0, we know by Lemma 6.7(a) that there exist b, c ∈ Mr such that b, c ≥ 0, kba∗ab − ck < , (6.9) r 2 and ϕ(b) = kbkr = 1 for every (not necessarily pure) state ϕ of Mr such that ϕ ◦ ιr ∗ q(u) u ∗ factors through Cr Gq(u) ; Eu . Let (eλ)λ∈Λ be an approximate identity for Cr (G; E), and ∗ ∗ for each ϕ ∈ Su, let Nϕ := {f ∈ Cr (G; E): ϕ(f f) = 0} be the null space for ϕ. Since the ∗ GNS representation πϕ satisﬁes πϕ(a) = 0, we have πϕ(ba ab) = 0, and so by the GNS construction, we have ∗ ∗ ϕ(ba ab) = lim πϕ(ba ab)(eλ + Nϕ) | eλ + Nϕ = 0, for each ϕ ∈ Su. (6.10) λ∈Λ

Together, equations (6.9) and (6.10) imply that for all ϕ ∈ Su, |ϕ(c)| = |ϕ(c)| ≤ |ϕ(c) − ϕ(ba∗ab)| + |ϕ(ba∗ab)| ≤ kc − ba∗abk < . (6.11) r 2 r ∗ G E ∗ G E ∗ q(u) u Let Qu : Cr (I ; I ) → Cr (Iq(u); Iu ) = Cr Gq(u) ; Eu be the surjective homomorphism of r −1 Corollary 4.8(a). Since c is a positive element of Mr, Qu(ιr (c)) is a positive element of ∗ q(u) u ∗ q(u) u Cr Gq(u) ; Eu , and so by [9, Proposition II.6.3.3], there is a pure state φ of Cr Gq(u) ; Eu satisfying r −1 r −1 φ Qu(ιr (c)) = Qu(ιr (c)) r . (6.12) r Deﬁne ψ : Mr → C by ψ(ιr(h)) := φ(Qu(h)). By Lemma 6.8, ψ is a pure state of Mr and ∗ G E ∗ q(u) u ψ ◦ ιr is a pure state of Cr (I ; I ) that factors through Cr Gq(u) ; Eu . Hence ψ ∈ Su, and so equations (6.12) and (6.11) imply that

r −1 r −1 Q (ι (c)) = φ Q (ι (c)) = |ψ(c)| < . (6.13) u r r u r 2 ∗ (0) −1 (0) Let evu be the evaluation map f 7→ f(u) on Cr G ; q (G ) , and let ρu := evu ◦ (Φr|Mr ). ∗ G E By parts (a) and (b) of Lemma 6.9, ρu is a state of Mr and ρu ◦ ιr is a state of Cr (I ; I ) ∗ q(u) u ∗ q(u) u that factors through Cr Gq(u) ; Eu . Hence there is a state κu of Cr Gq(u) ; Eu such that r ρu ◦ ιr = κu ◦ Qu. Thus, using equation (6.13) for the ﬁnal inequality, we obtain r −1 r −1 |ρu(c)| = κu Q (ι (c)) ≤ Q (ι (c)) < . (6.14) u r u r r 2 ∗ By Lemma 6.9(c), ρu := evu ◦ Φr is the unique state extension of ρu to Cr (G; E). By our choice of b, we have ρu(b) = ρu(b) = kbkr = 1. Thus, recalling the notation deﬁned in

Notation 5.1, we have b ∈ Uρu , and so Lemma 5.5 implies that b ∈ Mρu . Therefore, ∗ ∗ ∗ ρu(ba ab) = ρu(b) ρu(a a) ρu(b) = ρu(a a). (6.15) Using equation (6.15) for the second equality, we obtain ∗ ∗ ∗ Φr(a a)(u) = |ρu(a a)| = |ρu(ba ab)| ∗ ∗ ≤ |ρu(ba ab) − ρu(c)| + |ρu(c)| ≤ kba ab − ckr + |ρu(c)|. (6.16) 30 BECKY ARMSTRONG

Together, equations (6.16), (6.9), and (6.14) imply that Φ (a∗a)(u) ≤ kba∗ab − ck + |ρ (c)| < + = , r r u 2 2 ∗ which contradicts the inequality (6.8). Thus, we deduce that Φr(a a) = 0, which completes the proof. We conclude with a characterisation of simplicity of reduced twisted C*-algebras of effective Hausdorff étalegroupoids. Note that if the groupoid G is not effective, then ∗ characterising simplicity of Cr (G; E) in terms of G and E is a much harder problem; see [26, Remark 8.3]. Corollary 6.10. Let (E, i, q) be a twist over an effective Hausdorff étalegroupoid G. Then ∗ Cr (G; E) is simple if and only if G is minimal. Proof. Suppose that G is not minimal. Then there exists a nonempty proper closed (0) −1 invariant subset K of G . Define GK := s (K). Since K is closed and invariant, (0) GK is a closed étalesubgroupoid of G with unit space GK = K. Hence Proposi- −1 −1 tion 2.10 implies that q (GK ) is a twist over GK . The restriction map f 7→ f|q (GK ) −1 is a ∗-homomorphism from Σc(G; E) to Σc GK ; q (GK ) , and so it extends to a C*- ∗ ∗ −1 homomorphism resK : Cr (G; E) → Cr GK ; q (GK ) . Since GK is nonempty, resK is not the zero map, and since GK 6= G, resK is not injective. Hence ker(resK) is a nonzero proper ∗ ∗ ideal of Cr (G; E), and so Cr (G; E) is not simple. For the converse, suppose that G is minimal. Let Dr denote the completion of the set (0) D0 := f ∈ Σc(G; E): q(supp(f)) ⊆ G with respect to the reduced norm. Since G is G (0) E −1 (0) ∗ (0) −1 (0) ∗ effective, we have I = G and I = q (G ). Let ιr : Cr G ; q (G ) → Cr (G; E) ∗ (0) −1 (0) be the injective homomorphism of Proposition 6.1. Then ιr Cr G ; q (G ) = Dr. ∗ ∗ Let I be a nonzero ideal of Cr (G; E). Then there is a C*-homomorphism Ψ of Cr (G; E) such that I = ker(Ψ). Since I is nonzero, Ψ is not injective, and hence Theorem 6.4 implies that Ψ ◦ ιr is not injective either. Thus J := ker(Ψ ◦ ιr) is a nonzero ideal of ∗ (0) −1 (0) ∗ Cr G ; q (G ) , and we have ιr(J) = I ∩ Dr. To see that Cr (G; E) is simple, we must ∗ show that I = Cr (G; E). We know by [33, Theorem 5.2] that Dr contains an approximate ∗ identity for Cr (G; E) (see also, [35, Proposition 11.1.14]), and so it suffices to show that ∗ ιr(J) = Dr, because then Dr ⊆ I, and it follows that I = Cr (G; E). (0) Recall from [35, Theorem 11.1.11] that there is an isomorphism Υ: C0(G ) → Dr such (0) (0) that Υ(f)(i(x, z)) = z f(x) for all f ∈ C0(G ) and (x, z) ∈ G × T. Define (0) −1 F := {x ∈ G : f(x) = 0 for all f ∈ Υ (ιr(J))}. −1 (0) (0) Since Υ (ιr(J)) is a nonzero ideal of C0(G ), F is a proper closed subset of G , and −1 (0) Υ (ιr(J)) = {f ∈ C0(G ): f(x) = 0 for all x ∈ F }. −1 (0) To see that ιr(J) = Dr, we will prove the equivalent statement that Υ (ιr(J)) = C0(G ). Suppose that F 6= ∅. We will derive a contradiction by showing that F = G(0). Since G is minimal, the only closed invariant subsets of G(0) are ∅ and G(0) itself, and so it suffices to show that F is invariant. For this, fix x ∈ F , and suppose that γ ∈ G satisfies −1 s(γ) = x. We must show that r(γ) ∈ F . For this, fix f ∈ Υ (ιr(J)). We must show that f(r(γ)) = 0. Use Lemma 2.5 to choose a local trivialisation (Bα,Pα, φPα )α∈G of E such that each Bα is a bisection of G. Use Urysohn’s lemma to choose h ∈ Cc(G) such −1 that supp(h) ⊆ Bγ and h(γ) = 1. Recall from Lemma 2.4(c) that for each ε ∈ q (Bγ), there is a unique zε ∈ T such that ε = φPγ (q(ε), zε), and the map ε 7→ zε is continuous on −1 −1 q (Bγ). Thus ε 7→ zε h(q(ε)) is a continuous map from q (Bγ) to C. Define g : E → C A UNIQUENESS THEOREM FOR TWISTED GROUPOID C*-ALGEBRAS 31 by ( z h(q(ε)) if ε ∈ q−1(B ) : ε γ g(ε) = −1 0 if ε∈ / q (Bγ).

Since Pγ(γ) = φPγ (γ, 1), we have g(Pγ(γ)) = h(γ) = 1. By a similar argument to the one G E used in the proof of Lemma 4.6 to show that jε ∈ Σc(I ; I ), we see that g ∈ Σc(G; E). −1 −1 (0) Since supp(g) ⊆ q (Bγ) and supp(Υ(f)) ⊆ q (G ), it follows from equation (2.4) and the fact that Bγ is a bisection that ∗ −1 (0) (0) q supp g Υ(f) g ⊆ Bγ G Bγ = s(Bγ) ⊆ G , ∗ ∗ and thus g Υ(f) g ∈ Dr. Since Υ(f) is an element of the ideal I, it follows that g Υ(f) g ∈ I ∩ Dr = ιr(J). Therefore, since x ∈ F , we have g∗ Υ(f) g(i(x, 1)) = Υ−1g∗ Υ(f) g(x) = 0. (6.17) −1 Observe that i(x, 1) = s Pγ(γ) = Pγ(γ) i(r(γ), 1) Pγ(γ). Thus, since q(supp(g)) is contained in the bisection Bγ, equation (2.4) implies that ∗ ∗ −1 g Υ(f) g (i(x, 1)) = g Pγ(γ) Υ(f) i(r(γ), 1) g Pγ(γ) = f(r(γ)). (6.18) Together, equations (6.17) and (6.18) imply that f(r(γ)) = 0, and so F is invariant. Thus (0) −1 (0) F = G , which is a contradiction, because Υ (ιr(J)) is a nonzero ideal of C0(G ). −1 (0) Therefore, we must have F = ∅, and so Υ (ιr(J)) = C0(G ), as required.

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(B. Armstrong) Mathematical Institute, WWU Munster,¨ Einsteinstr. 62, 48149 Munster,¨ GERMANY Email address: [email protected]