Reconstruction of twisted Steinberg algebras
Becky Armstrong, Gilles G. de Castro, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick, Jacqui Ramagge, Aidan Sims, Benjamin Steinberg
aa r X i v : . [ m a t h . R A ] J a n RECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS
BECKY ARMSTRONG, GILLES G. DE CASTRO, LISA ORLOFF CLARK, KRISTIN COURTNEY,YING-FEN LIN, KATHRYN MCCORMICK, JACQUI RAMAGGE, AIDAN SIMS,AND BENJAMIN STEINBERG
Abstract.
We show how to recover a discrete twist over an ample Hausdorff groupoidfrom a pair consisting of an algebra and what we call a quasi-Cartan subalgebra . Weidentify precisely which twists arise in this way (namely, those that satisfy the localbisection hypothesis ), and we prove that the assignment of twisted Steinberg algebrasto such twists and our construction of a twist from a quasi-Cartan pair are mutuallyinverse. We identify the algebraic pairs that correspond to effective groupoids and toprincipal groupoids. We also indicate the scope of our results by identifying large classesof twists for which the local bisection hypothesis holds automatically.
Contents
1. Introduction 2Related work 3Outline of the paper 32. Preliminaries 52.1. Abelian R -algebras without torsion 52.2. Groupoids 62.3. Discrete twists 72.4. Twisted Steinberg algebras 92.5. Normalisers and inverse semigroups 133. Algebraic quasi-Cartan pairs 144. Building an algebraic quasi-Cartan pair from a twist 185. Building a twist from a pair of algebras 245.1. The groupoid Σ 245.2. The groupoid G A ∼ = A R ( G ; Σ) 297. Algebraic information from the isotropy structure of G Date : 22nd January 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Steinberg algebra, groupoid, cohomology.This research was supported by: the project-oriented workshop “Women in Operator Algebras” (18w5168)in November 2018, which was funded and hosted by the Banff International Research Station; the Aus-tralian Research Council Discovery Project DP200100155; the Sydney Mathematical Research InstituteDomestic Visitor Program; CAPES-PrInt grant number 88887.368595/2019-00; the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2044 –390685587, Mathematics M¨unster – Dynamics – Geometry – Structure; the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442; ERC Advanced Grant834267 - AMAREC; and PSC-CUNY. Introduction
In this paper we provide a concrete representation theorem for a large class of abstractalgebras by establishing a correspondence between algebraic twists over ample Hausdorffgroupoids and a corresponding class of pairs (
A, B ) of abstract algebras.When trying to understand an algebra, it is often helpful to describe it in terms of a con-crete representation. For example: Stone duality for commutative algebras generated byidempotents; the duality between commutative von Neumann algebras and hyperstoneanspaces; Gelfand duality for commutative C*-algebras; and the Gelfand–Naimark theoremfor noncommutative C*-algebras.A more recent example comes from Leavitt path algebras, whose advent [1, 3] hassparked substantial activity around interactions between abstract algebra and C*-algebrasover the last decade or so. It was discovered early on that the complex Leavitt path al-gebra of a graph embeds in the graph C*-algebra, providing a concrete representation ofany given Leavitt path algebra by bounded operators on a Hilbert space. But a clearunderstanding of the striking structural similarities between Leavitt path algebras overgeneral rings and the corresponding graph C*-algebras was only achieved through thedevelopment of convolution algebras of functions on groupoids [45, 15], now called Stein-berg algebras, as a unifying framework. It emerged that both Leavitt path algebrasand graph C*-algebras can be realised as algebras of functions on the underlying graphgroupoid [15]. Moreover, the groupoid can be recovered from either the graph C*-algebratogether with its abelian subalgebra generated by the range projections of its generat-ing partial isometries [11], or from the Leavitt path algebra (over a very broad class ofrings) and its corresponding abelian subalgebra [10, 12, 47]. This led to deep connectionsbetween abstract algebra and symbolic dynamics [13], and these have been significantlyextended through recent reconstruction results that show how increasingly broad classesof groupoids can be reconstructed both from their C*-algebras [43, 14] and from theirSteinberg algebras [2, 47]. Analogously to the case for groupoid C*-algebras, there isa disintegration theorem [46] for realising modules over Steinberg algebras as sheavesover the groupoid, and Morita equivalence of Steinberg algebras can often be seen at thegroupoid level [17, 46].The upshot is that there are significant advantages to being able to recognise a givenalgebra as the Steinberg algebra, or a twisted Steinberg algebra [4], of a groupoid. Thusit is important to understand exactly which algebras admit such a model.This question was first answered for von Neumann algebras by Feldman and Moore[19, 20, 21]. They showed that every Borel 2-cocycle on a Borel equivalence relation givesrise to a von Neumann algebra and a so-called Cartan subalgebra (roughly, an abeliansubalgebra whose normaliser generates the whole algebra); and that the equivalence re-lation and 2-cocycle can be recovered (up to cohomology) from the pair of algebras. Acorresponding theorem for C*-algebras was developed by Renault and by Kumjian firstfor principal groupoids [25] and then for effective groupoids [43], using Kumjian’s notionof a twist over an ´etale groupoid in place of a 2-cocycle. It was these results that Mat-sumoto and Matui subsequently used [32] to prove that flow equivalence of shift spacesis characterised by diagonal-preserving isomorphism of the associated Cuntz–Krieger al-gebras, leading to the surge of activity around reconstruction of groupoids that are noteven effective from the C*-algebras discussed above.Renault’s results beg an extension to abstract algebras, following the program of “al-gebraisation” of operator-theoretic ideas going back to von Neumann and Kaplansky (see[23]) described in [16]: in the words of Berberian [6] “if all the functional analysis is
ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 3 stripped away . . . what remains should stand firmly as a substantial piece of algebra, com-pletely accessible through algebraic avenues.” The groundwork for such an extension waslaid recently in [4], where a notion of an algebraic twist over an ample groupoid, suit-able for constructing twisted Steinberg algebras, was developed, and the structure of theresulting twisted Steinberg algebras was analysed.In this paper we provide such an extension. We identify a class of abelian subalge-bras, called algebraic quasi-Cartan subalgebras , in abstract algebras over indecomposablecommutative rings. Our main theorems establish a correspondence between algebraicquasi-Cartan pairs and twists over ample Hausdorff ´etale groupoids that satisfy a localbisection hypothesis (analogous to that of [47]) that generalises the no-nontrivial-unitscondition for twisted group rings. Indeed, we show that the local bisection hypothesisfollows from the no-nontrivial-units condition for sufficiently many of the twisted grouprings over the interior of the isotropy of the groupoid. This guarantees that our resultsapply to many important examples, as we detail in the final section of the paper—carryingour results for algebras beyond the current state of the art for C*-algebras. We also pickout from amongst all algebraic quasi-Cartan pairs the algebraic Cartan pairs that cor-respond to the effective groupoids of Renault’s setting, and the algebraic diagonal pairsthat correspond to the principal groupoids of Kumjian’s setting.This opens interesting new avenues of study for algebras admitting quasi-Cartan sub-algebras, and provides a wealth of invariants for algebraic quasi-Cartan pairs: for exam-ple, the topological full-group of the underlying groupoid G , with intriguing links to theThompson groups [39, 33]; the homology groups of G , which have promising connectionsto computations of K -theory in the C*-algebraic setting [34, 35, 36]; and the twist itself,which behaves like a second cohomology class for the orbit space of the groupoid that canbe used as a classifying invariant for Fell algebras [22]. Related work.
In the final stages of preparation of this manuscript we became aware ofBice’s independent work [7] on ringoid bundles, and in particular on Steinberg rings andSteinberg bundles. We thank Bice bringing his work to our attention, and for sharing hispreprint.Bice’s results deal, at their most general, with topological categories appropriately fi-bred over ´etale groupoids. His main results are about
Steinberg bundles which, looselyspeaking, bear the same relationship to our discrete R -twists as Fell bundles bear totwists in the C*-algebraic setting; in particular, in Bice’s setting, the fibres of q − ( G ) canvary, and need not be commutative. Bice shows that every Steinberg bundle determinesa Steinberg ring : a ring A with distinguished subsemigroups Z ⊆ S ⊆ A (correspondingto I ( B ) ⊆ N ( B ) ⊆ A ), and a suitable notion of a conditional expectation Φ : A → S such that Z ⊆ Φ( A ), satisfying an additional support condition. He demonstrates thatevery Steinberg ring determines a corresponding Steinberg bundle (via an ultrafilter con-struction) whose Steinberg ring is the original ( A, S, Z,
Φ). However, the same pair (
A, Z )could admit multiple Steinberg-ring structures, yielding different Steinberg bundles.Our set-up is less general, but requires less algebraic data to describe the algebraicobjects of study: the quasi-Cartan pair (
A, B ) is all the information required, and then N ( B ) and P can be recovered. We then obtain a complete and concrete correspondencebetween quasi-Cartan pairs and R -twists satisfying the local bisection hypothesis. Outline of the paper.
The paper is organised as follows. In Section 2, we introducepreliminaries on abelian R -algebras without torsion (this is a standing assumption on theabelian subalgebras B in our quasi-Cartan pairs), on groupoids and discrete R -twists, ontwisted Steinberg algebras, and on inverse semigroups of normalisers of abelian subalge-bras. B. ARMSTRONG ET AL.
In Section 3, we introduce our three notions of algebraic pairs (see Definition 3.3), andwe prove in Lemma 3.5 that every algebraic diagonal pair is an algebraic Cartan pair.We also show that for a given quasi-Cartan pair (
A, B ), there is only one conditionalexpectation from A to B that satisfies a key defining algebraic condition (AQP). To finishthe section, we show that if ( A, B ) is an algebraic Cartan pair, then there is a uniqueconditional expectation from A to B , and we show in Lemma 3.6 that every algebraicCartan pair is an algebraic quasi-Cartan pair.In Section 4, we study the pairs of algebras arising from discrete R -twists, and demon-strate that such a pair is an algebraic quasi-Cartan pair if and only if the twist satisfiesthe local bisection hypothesis. We also provide tools for checking the local bisection hy-pothesis, and we finish the section by showing in Proposition 4.8 that twists over principalgroupoids yield algebraic diagonal pairs, and twists over effective groupoids yield algebraicCartan pairs.In Section 5, we show how to construct a twist from a pair of algebras. We havemade an effort to be clear about which hypotheses are required for each part of theconstruction. In Section 5.1, we begin with a pair ( A, B ) such that B is without torsionand the idempotents of B are a set of local units for A , and we describe an associatedgroupoid Σ of ultrafilters of normalisers of B and some of its key properties. In Section 5.2we additionally assume that B is the R -linear span of its idempotent elements. With thishypothesis, we construct a quotient G of Σ by the action of R × , and show that G is anample groupoid. In Section 5.3, we show that under the same hypotheses we obtain asequence G (0) × R × ֒ → Σ ։ G that has all of the properties of a discrete R -twist over G except, potentially, Hausdorffness. We show that if ( A, B ) is an algebraic quasi-Cartanpair, then G (and hence also Σ) is indeed Hausdorff—this is all contained in Theorem 5.6and Proposition 5.10.In Section 6, we prove our first main result. We start with an algebraic quasi-Cartanpair ( A, B ), and consider the twist Σ over G constructed in the preceding section. Ourmain theorem, Theorem 6.6, shows that there is an explicit isomorphism of A onto thetwisted Steinberg algebra A R ( G ; Σ) that restricts to the standard isomorphism of B onto A R ( G (0) ; q − ( G (0) )) ∼ = A R ( G (0) ) arising from Stone duality as in [24, Th´eor`eme 1].In Section 7, we explore the relationship between properties of a twist Σ over G sat-isfying the local bisection hypothesis and properties of the associated Steinberg algebra.Specifically, we prove that if ( A, B ) is an algebraic quasi-Cartan pair, then it is a Cartanpair if and only if the associated groupoid G is effective (so that our terminology is con-sistent with Renault’s in [43]) and that it is a diagonal pair if and only if G is principal(so that our terminology is consistent with Kumjian’s in [25]).We see in Section 6 that passing from an algebraic quasi-Cartan pair to the associatedtwist and then to the associated twisted Steinberg algebra and its diagonal subalgebrarecovers the original pair. In Section 8 we consider the dual approach where we startwith a twist, pass to the associated pair of algebras, and then construct the associatedsequence of groupoids. We prove in Proposition 8.4 that given any twist Σ over an ampleHausdorff groupoid G , if the algebraic pair ( A, B ) consists of a Steinberg algebra anda diagonal subalgebra, then the groupoid Σ ′ of ultrafilters of normalisers of B admits acontinuous open embedding of Σ that restricts to a homeomorphism of unit spaces. Thenin Theorem 8.7, we show that this embedding is an isomorphism if and only if ( A, B ) is analgebraic quasi-Cartan pair (equivalently, if and only if Σ → G satisfies the local bisectionhypothesis). We deduce in Corollary 8.10 that given twists Σ → G and Σ → G suchthat Σ → G satisfies the local bisection hypothesis, the twists are isomorphic (whichimplies in particular that Σ → G also satisfies the local bisection hypothesis) if andonly if the associated algebraic pairs are isomorphic. We then deduce in Corollary 8.11 ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 5 that if G is effective, then the twists are isomorphic (which implies in particular that G is effective) if and only if there is an isomorphism A R ( G ; Σ ) A R ( G ; Σ ) of twistedSteinberg algebras that carries the diagonal subalgebra of A R ( G ; Σ ) into the diagonalsubalgebra of A R ( G ; Σ ).Finally, in Section 9 we demonstrate the applicability of our results by showing thatthere are substantial classes of twists that satisfy the local bisection hypothesis, andtherefore correspond to algebraic quasi-Cartan pairs. We consider groups H with the unique product property ; that is, that products AB of finite subsets of H always containan element with a unique factorisation of the form ab with a ∈ A and b ∈ B . (Examplesinclude right-orderable groups, and in particular torsion-free abelian groups.) By adaptingthe approach used in [47] for untwisted group rings, we show that if R is reduced andindecomposable and H is a group with the unique product property, then every twistedgroup R -algebra of H has no nontrivial units. Combining this with our results in Section 4,we deduce that for reduced and indecomposable R , all R -twists over groupoids G suchthat the fibres of the interior of the isotropy of G have the unique product property satisfythe local bisection hypothesis (the reduced assumption can be dropped if the groupoid iseffective). In particular, this includes all twists over ample Deaconu–Renault groupoids;we give a specific application to twisted Kumjian–Pask algebras.2. Preliminaries
Abelian R -algebras without torsion. Throughout this article, R denotes a com-mutative ring with identity, R × is the group of units of R , and A is an R -algebra. Wealways assume that B ⊆ A is a commutative (also called abelian) subalgebra with idem-potents I ( B ). We also ask that B is without torsion with respect to R in the sense usedin [24]; specifically,if e ∈ I ( B ) \{ } and t ∈ R satisfy te = 0 , then t = 0 . (WT)Condition (WT) holds automatically if R is a field (and also if R is an integral domainand B is a torsion-free R -module, as explained below). Observe that if B has at leastone nonzero idempotent, then condition (WT) also implies that R is indecomposable in the sense that its only idempotents are 0 and 1: if t ∈ R satisfies t = t , then t e = te for any e ∈ I ( B ) \{ } , and then t (1 − t ) e = 0, forcing t = 0 or t = 1 (since(1 − t ) e ∈ I ( B )). Since we will be working exclusively with algebras B that are generatedby their idempotent elements, it follows that R is generally indecomposable. This is alsothe condition on R used in [47]. The condition of a ring R being indecomposable is anatural one in commutative ring theory. For instance, recall that the Zariski (or prime)spectrum Spec( R ) of R is the set of prime ideals of R with the topology whose basic opensubsets are of the form D ( r ) = { p ∈ Spec( R ) : r / ∈ p } with r ∈ R . It is well known thatSpec( R ) is a connected space if and only if R is indecomposable.We will frequently use without comment the following observation. If B is spanned asan R -module by I ( B ), then each element b ∈ B can be written as b = P ni =1 t i e i with e , . . . , e n ∈ I ( B ) mutually orthogonal and t , . . . , t n ∈ R . Indeed, if b = P e ∈ F t e e with F ⊆ I ( B ) finite, then F generates a finite Boolean algebra, where we recall that the joinof commuting idempotents e and f is e ∨ f = e + f − ef , their meet is e ∧ f = ef , andtheir relative complement is e \ f = e − ef . We can then take e , . . . , e n to be the atoms(minimal idempotents) of this Boolean algebra.An R -module M is called torsion-free if rm = 0 implies that either m = 0 or r is azero divisor, for r ∈ R and m ∈ M . Notice that if I ( B ) generates B as an R -algebra, For us, indecomposable implicitly implies unital.
B. ARMSTRONG ET AL. then condition (WT) implies that B is a torsion-free R -module. Indeed, if 0 = b ∈ B and tb = 0, we can write b = P ni =1 t i e i with 0 = t i ∈ R and e i ∈ I ( B ) with e , . . . , e n mutuallyorthogonal. Then 0 = tbe = tt e , and so tt = 0 by (WT), whence t is a zero divisor. Inparticular, if R is an integral domain and B is generated by I ( B ), then B is torsion-freeif and only if it satisfies condition (WT).2.2. Groupoids.
We use G to mean a locally compact Hausdorff topological groupoidwith unit space G (0) , composable pairs G (2) ⊆ G × G , and range and source maps r , s : G → G (0) . We will refer to such groupoids as Hausdorff groupoids . We evaluate compositionof groupoid elements from right to left, which means that γγ − = r ( γ ) and γ − γ = s ( γ ),for all γ ∈ G .For each x ∈ G (0) , we define G x := s − ( x ) , G x := r − ( x ) , and G xx := G x ∩ G x . For any two subsets U and V of a groupoid G , we define U V := { αβ : ( α, β ) ∈ ( U × V ) ∩ G (2) } and U − := { α − : α ∈ U } . We call a subset B of G a bisection if there exists an open subset U of G such that B ⊆ U , and r | U and s | U are homeomorphisms onto open subsets of G (0) . We say that G is ´etale if r (or, equivalently, s ) is a local homeomorphism. If G is ´etale, then G (0) is open,and both G x and G x are discrete in the subspace topology for any x ∈ G (0) . We recallthat G is ´etale if and only if G has a basis of open bisections. We say that G is ample ifit has a basis of compact open bisections. If G is ´etale, then G is ample if and only if itsunit space G (0) is totally disconnected (see [18, Proposition 4.1]). It is well known thatan ´etale groupoid is Hausdorff if and only if its unit space is closed.If B and D are compact open bisections of an ample groupoid, then B − and BD arealso compact open bisections. The collection of compact open bisections is an inversesemigroup under these operations (see [42, Proposition 2.2.4]).The isotropy of a groupoid G is the setIso( G ) := { γ ∈ G : r ( γ ) = s ( γ ) } = [ x ∈ G (0) G xx . We say that G is principal if Iso( G ) = G (0) , and that G is effective if the topologicalinterior of Iso( G ) is equal to G (0) . So every principal ´etale groupoid is effective. Remark . In full generality, effectiveness should not be confused with several relatednotions. There is the notion of a groupoid being topologically principal, which meansthat points with trivial isotropy (that is, G xx = { x } ) are dense in G (0) , and the weakernotion of a groupoid being topologically free, which means that Iso( G ) \ G (0) has emptyinterior. Both effective and topologically principal ´etale groupoids are topologically free.For second-countable ´etale groupoids, the notions of topologically principal and topolog-ically free coincide, whereas for Hausdorff ´etale groupoids, being effective is equivalent tobeing topologically free. Hence for second-countable Hausdorff ´etale groupoids, all threenotions agree. However, there are examples of minimal ample groupoids showing thatthere are no other implications between these notions.For the majority of this paper, the groupoids we work with are ample Hausdorffgroupoids, but we will make these assumptions explicit throughout. ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 7
Discrete twists.
We next recall [4, Definition 4.1]. Let G be a Hausdorff ´etalegroupoid, let R be a commutative unital ring, and let T be a subgroup of R × . A discretetwist over G is a sequence G (0) × T i ֒ → Σ q ։ G, (2.1)where the groupoid G (0) × T is regarded as a trivial group bundle with fibres T , Σ is agroupoid with Σ (0) = i (cid:0) G (0) × { } (cid:1) , and i and q are continuous groupoid homomorphismsthat restrict to homeomorphisms of unit spaces, such that the following conditions hold:(DT1) The sequence is exact, in the sense that i ( { x } × T ) = q − ( x ) for every x ∈ G (0) , i is injective, and q is surjective.(DT2) The groupoid Σ is a locally trivial G -bundle, in the sense that for each α ∈ G , thereis an open bisection B α of G containing α , and a continuous map P α : B α → Σsuch that(i) q ◦ P α = id B α ;(ii) the map ( β, t ) i ( r ( β ) , t ) P α ( β ) is a homeomorphism from B α × T to q − ( B α ).(DT3) The image of i is central in Σ, in the sense that i ( r ( σ ) , t ) σ = σ i ( s ( σ ) , t ) for all σ ∈ Σ and t ∈ T .We denote a discrete twist over G by (Σ , i, q ), or by Σ → G , or simply by Σ. Weidentify Σ (0) with G (0) via i (or via q | Σ (0) ). Note that [4, Definition 4.1] also requires Σ tobe Hausdorff, but we show in Corollary 2.3 below that this is automatic.Recall from [4, Page 14] that there is a continuous free action of T on Σ given by t · σ = i ( r ( σ ) , t ) σ for all t ∈ T and σ ∈ Σ . In this paper, we will only be interested in the setting where T = R × . We will sometimesemphasise the dependence on R × by calling Σ a discrete R -twist over G , or sometimesjust an R -twist over G .We begin by observing that if Σ is a discrete twist, then it, too, is an ´etale groupoid,and the map q is a quotient map. In fact, we have the following characterisations ofdiscrete twists, some of which are easier to check than others. Proposition 2.2.
Consider an exact sequence of groupoids as in (2.1) with G ´etale butnot necessarily Hausdorff. Suppose that conditions (DT1) and (DT3) in the definition ofa discrete twist are satisfied. If q is a local homeomorphism, then Σ is ´etale. The followingare equivalent. (1) Condition (DT2) in the definition of a discrete twist is satisfied. (2)
The map i is open and q is a covering map. (3) The map i is open and the map q is a local homeomorphism. (4) The maps i and q are open, and Σ is ´etale.Proof. If q is a local homeomorphism, then since the source map for Σ is the composi-tion of q with the source map for G , and since local homeomorphisms are closed undercomposition, Σ is ´etale.(1) = ⇒ (2). Condition (DT2) says that Σ is a locally trivial fibre bundle with discretefibres T , and hence q is a covering map (and so a local homeomorphism). Thus Σ is ´etaleby the previous assertion. Now [4, Lemma 4.3(b)] (which formally assumes that Σ and G are Hausdorff but only uses that they are ´etale and that conditions (DT1)–(DT3) aresatisfied) implies that i is open.(2) = ⇒ (3) is immediate: every covering map is a local homeomorphism.(3) = ⇒ (4) is also immediate: if q is a local homeomorphism then it is open, and Σ is´etale by the first assertion of the proposition. B. ARMSTRONG ET AL. (4) = ⇒ (1). Suppose that i and q are open and Σ is ´etale. We show (DT2), that is,that Σ is locally trivial. Fix α ∈ G and σ ∈ Σ with q ( σ ) = α , and choose an openbisection V ⊆ Σ with σ ∈ V using that Σ is ´etale. Note that q | V is injective, because if q ( β ) = q ( β ′ ) with β, β ′ ∈ V , then s ( β ) = s ( β ′ ), and so β = β ′ since V is a bisection. Put B α := q ( V ) and P α := ( q | V ) − . Then P α is a homeomorphism since q is open by hypothesis.Moreover, B α is an open bisection, because V is an open bisection and q is an open mapthat respects r and s . It remains to show that the map φ P α : ( β, t ) i ( r ( α ) , t ) P α ( β ) isa homeomorphism from B α × T to q − ( B α ). Since i , r , P α , and the multiplication in an´etale groupoid are all continuous and open, it follows that φ P α is continuous and open.We now show that it is bijective. If i ( r ( β ) , t ) P α ( β ) = i ( r ( γ ) , t ′ ) P α ( γ ), then applying q to both sides shows that β = γ . We then obtain that i ( r ( β ) , t ) = i ( r ( β ) , t ′ ), and so t = t ′ by the injectivity of i . Finally, to see that φ P α is surjective, fix β ∈ q − ( B α ). Then q ( β ) = q ( P α ( q ( β ))), and so βP α ( q ( β )) − ∈ q − ( G (0) ). Thus β = i ( r ( β ) , t ) P α ( q ( β )) forsome t ∈ T . This completes the proof. (cid:3) The only time we apply Proposition 2.2 to non-Hausdorff groupoids is in the proof ofTheorem 5.6, where we only use the implication (4) = ⇒ (1), which does not rely on [4,Lemma 4.3(b)] (stated only for Hausdorff groupoids). Corollary 2.3.
Let (Σ , i, q ) be a discrete twist over a Hausdorff ´etale groupoid G . Then Σ is a Hausdorff ´etale groupoid, and q is a local homeomorphism. In particular, G hasthe quotient topology.Proof. The map q is a covering map, and hence a local homeomorphism, byProposition 2.2. Thus Σ is Hausdorff, because any covering space of a Hausdorff space isHausdorff. (cid:3) In [4, Lemma 4.3(c)] it is shown that if (Σ , i, q ) is a discrete twist, then the sets B α and maps P α of condition (DT2) can be chosen such that P α ( B α ∩ G (0) ) ⊆ Σ (0) . Acontinuous map P α : B α → Σ defined as in condition (DT2) that satisfies this additionalcondition is called a (continuous) local section . If G is ample, then the open bisectionsfrom condition (DT2) can be chosen to be compact.If (Σ , i, q ) is a discrete twist, then by definition, every element of G has an open bisectionneighbourhood that admits a local trivialisation. To finish this section, we observe thatthe reduction of Σ to any countable union of compact open subsets of G is topologicallytrivial. The proof of the following result is essentially the proof of [4, Theorem 4.10]. Lemma 2.4.
Let G be an ample Hausdorff groupoid, and let (Σ , i, q ) be a discrete R -twist over G . Suppose that U ⊆ G is a countable union of compact open subsets. Thenthere exists a continuous section S : U → Σ that trivialises Σ over U , in the sense that ( γ, t ) t · S ( γ ) is a homeomorphism of U × R × onto q − ( U ) . In fact, in Lemma 2.4 it is enough for U \ G (0) to be paracompact rather than U being acountable union of compact open sets, since it is well known that a paracompact locallycompact space can be partitioned into clopen σ -compact subspaces.In [4], the authors studied discrete twists arising from continuous 2-cocycles (see [4,Example 4.5]), and showed that every discrete R -twist (Σ , i, q ) admitting a continuousglobal section for q is induced by a continuous 2-cocycle (see [4, Proposition 4.8]). Acontinuous 2-cocycle on a Hausdorff ´etale groupoid G is a continuous map σ : G (2) → R × that satisfies the 2 -cocycle identity : σ ( α, β ) σ ( αβ, γ ) = σ ( α, βγ ) σ ( β, γ ) for all composable α, β, γ ∈ G ; and is normalised , in the sense that σ ( r ( γ ) , γ ) = σ ( γ, s ( γ )) = 1 for all γ ∈ G . Example . We now sketch an argument showing that there are discrete R -twists overprincipal ample groupoids that do not admit a continuous global section, and hence are ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 9 not equivalent to a twist coming from a continuous 2-cocycle, for a fairly broad class ofrings R . Kumjian shows in [25, Proposition 4.4] that if X is a locally compact spaceand G is the principal groupoid corresponding to the equivalence relation X × X , thentwists (in the classical sense) over G are in bijection with principal circle bundles over X . Moreover, it is observed in [37] that a twist over G admits a continuous globalsection if and only if the corresponding circle bundle is trivial. The same constructioncan be applied mutatis mutandis in our setting to show that if X is a totally disconnectedlocally compact Hausdorff space, then discrete R -twists over X × X are in bijection withprincipal R × -bundles over X (where R × is viewed as a discrete abelian group), and thetwist admits a continuous global section if and only if the corresponding bundle is trivial.The construction just relies on the fact that for a suitable topological abelian group A ,the isomorphism classes of principal A -bundles over X form an abelian group under the“Baer sum”, with inverse given by inverting the action. (In particular, one needs toreplace complex conjugation by inversion in the construction given in [25].)So it suffices to give an example of a totally disconnected locally compact Hausdorffspace X that admits a nontrivial R × -bundle for some ring R . We will show that this canbe done for a ring R whose group of units is isomorphic to Z / Z × A for some abeliangroup A . This includes Z , and more generally any ordered ring (for example, a subringof an ordered field like R ), and finite fields of order congruent to 3 modulo 4.In [50, Example 5.3], a totally disconnected locally compact Hausdorff space X is givenwith a covering U by compact open sets such that the first ˇCech cohomology group of X with respect to this covering and with coefficients in some sheaf of abelian groups does notvanish. In the footnote to the example, the author points out that one can, in fact, provethat the first cohomology group H ( U ; Z / Z ) with coefficients in the constant sheaf withvalue Z / Z does not vanish (with respect to this covering or any covering by compactopen sets); some further details are provided in [44]. By the theory of principal bundles,this is equivalent to the existence of a nontrivial principal ( Z / Z )-bundle over X . In fact,if R × ∼ = Z / Z × A for an abelian group A , then since Z / Z is a retract of R × , we see that H ( U ; Z / Z ) is a retract of H ( U ; R × ) and hence the latter is nontrivial. Therefore, thespace X admits a nontrivial principal R × -bundle for any such ring R .This then shows that the principal groupoid X × X admits a discrete R -twist ( R asabove) with no continuous global sections; that is, a twist that does not come from acontinuous 2-cocycle.2.4. Twisted Steinberg algebras.
In [4], the authors defined the twisted Steinbergalgebra arising from a discrete R -twist (Σ , i, q ) over G . This was accomplished in thesetting where Σ is topologically trivial (that is, Σ admits a continuous global section for q , and is thus induced by a continuous 2-cocycle). The original definition also allowed fortwists over groupoids with fibres a subgroup T ≤ R × . We include the definition whichwe use below.Given a topological space X and a ring R , we write C ( X, R ) for the R -module of locallyconstant maps from X to R . For f ∈ C ( X, R ), we definesupp( f ) := { x ∈ X : f ( x ) = 0 } , which is a clopen set. We also define C c ( X, R ) := { f ∈ C ( X, R ) : supp( f ) is compact } , which is an R -submodule of C ( X, R ). Definition 2.6.
Let G be an ample Hausdorff groupoid, and let (Σ , i, q ) be a discrete R -twist over G . We say that f ∈ C (Σ , R ) is R × -contravariant if f ( t · σ ) = t − f ( σ ) for all t ∈ R × and σ ∈ Σ, and we define A R ( G ; Σ) := { f ∈ C (Σ , R ) : f is R × -contravariant and q (supp( f )) is compact } . As a notational convenience, given a discrete R -twist (Σ , i, q ) over G , for f ∈ A R ( G ; Σ),we define supp G ( f ) := q (supp( f )) ⊆ G. We then have supp( f ) = q − (supp G ( f )) because of the R × -contravariance of f . Since q is a quotient map and supp( f ) is clopen, supp G ( f ) is also clopen. Thus the condi-tion that q (supp( f )) is compact and the condition that q (supp( f )) is compact (as in [4,Definition 4.17]) are equivalent.In [4], the authors were mainly concerned with topologically trivial twists, and in partic-ular proved that A R ( G ; Σ) has a well-defined multiplication under this assumption. Herewe show that A R ( G ; Σ) is an R -algebra even when the twist is not topologically trivial.We now establish that elements of A R ( G ; Σ) can be expressed in terms of their restric-tion to any “slice” of their support with respect to the action of R × on Σ. Lemma 2.7.
Let G be an ample Hausdorff groupoid and let (Σ , i, q ) be a discrete twistover G . Suppose that X is an open subset of Σ such that q | X is injective. Let f : X → R be a locally constant compactly supported function. Then there is a unique element ˜ f of A R ( G ; Σ) such that supp( ˜ f ) ⊆ R × · X and ˜ f | X = f .Proof. We have q ( t · σ ) = q ( σ ) for all t ∈ R × and σ ∈ Σ, and so since q | X is injective, X intersects each R × -orbit at no more than one point. So there is a well-defined function f ′ : R × · supp( f ) → R satisfying f ′ ( t · σ ) = t − f ( σ ) for all σ ∈ supp( f ) and t ∈ R × .Since f is locally constant and σ t · σ is a homeomorphism for each fixed t ∈ R × ,the function f ′ is also locally constant. It follows from q being an open map, q (supp( f ))being compact, and G being Hausdorff that q (supp( f )) is clopen, and hence R × · supp( f )is also clopen. Therefore, the function ˜ f given by ˜ f | R × · supp( f ) = f ′ and ˜ f | Σ \ R × · supp( f ) = 0is locally constant. Since q (supp( ˜ f )) = q (supp( f )), we have that q (supp( ˜ f )) is compact,and ˜ f is R × -contravariant by definition. The uniqueness is clear: if g ∈ A R ( G ; Σ) agreeswith f on X then it agrees with f ′ on R × · X , and so if in addition supp( g ) ⊆ R × · X ,then g = ˜ f . (cid:3) Suppose that X ⊆ Σ is a compact open bisection. Then in particular, q | X is injectivebecause q respects r and s and restricts to a homeomorphism of unit spaces. We willdenote by ˜1 X the element of A R ( G ; Σ) obtained by applying Lemma 2.7 to the constantfunction 1 X : X → R that maps every element of X to 1. Proposition 2.8.
Let G be an ample Hausdorff groupoid, and let (Σ , i, q ) be a discrete R -twist. Then for each f ∈ A R ( G ; Σ) , there exist a finite set F of compact open bisectionsof Σ with mutually disjoint images in G , and elements r U of R , for each U ∈ F , suchthat f = P U ∈F r U ˜1 U .Proof. We apply Lemma 2.4 to find a section S : supp G ( f ) → supp( f ) that trivialisesΣ over supp G ( f ). Then X := S (supp G ( f )) ∼ = supp G ( f ) × { } is compact and open,and f | X belongs to the (untwisted) Steinberg algebra A R (Σ). Consequently, by [45,Proposition 4.14], we can write f | X = P U ∈F r U U as a finite R -linear combination of In [4, Definition 4.17], the authors consider R × -equivariant functions f ∈ C (Σ , R ), which satisfy f ( t · σ ) = tf ( σ ) for all t ∈ R × and σ ∈ Σ. We remark that the results in [4, Section 4.3] go through withDefinition 2.6 as well. See [4, Remark 4.24]. Our use of the term “contravariant” is not related to thenotion of a “contravariant functor”.
ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 11 characteristic functions of mutually disjoint compact open bisections. Since q is injectiveon X , the images of the elements of F under q are also mutually disjoint.By Lemma 2.7, each ˜1 U belongs to A R ( G ; Σ), and so P U ∈F r U ˜1 U ∈ A R ( G ; Σ) because A R ( G ; Σ) is an R -module. Now, since the functions f and P U ∈F r U ˜1 U agree on X andboth belong to A R ( G ; Σ) with support contained in supp( f ) = R × · X , Lemma 2.7 impliesthat f = P U ∈F r U ˜1 U . (cid:3) Proposition 2.9.
Let G be an ample Hausdorff groupoid, and let (Σ , i, q ) be a discrete R -twist over G . Given f, g ∈ A R ( G ; Σ) , let S : supp G ( f ) → supp( f ) be any section (notnecessarily continuous) for q on supp G ( f ) . The formula ( f ∗ g )( σ ) := X α ∈ G r ( σ ) ∩ supp G ( f ) f ( S ( α )) g ( S ( α ) − σ ) does not depend on the choice of S . This formula defines an associative multiplication on A R ( G ; Σ) , making it into an R -algebra.Proof. It is straightforward to check that A R ( G ; Σ) is an R -module. Fix f, g ∈ A R ( G ; Σ).For each σ ∈ Σ, define F r ( σ ) := G r ( σ ) ∩ supp G ( f ). Since each F r ( σ ) is compact anddiscrete, and hence finite, the convolution formula makes sense.Suppose that S and S ′ are two sections for q on supp G ( f ). Then there is a unique (notnecessarily continuous) function α t α from supp G ( f ) to R × such that S ( α ) = t α · S ′ ( α )for all α ∈ supp G ( f ). Fix σ ∈ Σ. Using the centrality of i ( G (0) × R × ) and the R × -contravariance of f and g for the second equality, we obtain X α ∈ F r ( σ ) f ( S ( α )) g ( S ( α ) − σ ) = X α ∈ F r ( σ ) f ( t α · S ′ ( α )) g (( t α · S ′ ( α )) − σ )= X α ∈ F r ( σ ) t − α f ( S ′ ( α )) t α g ( S ′ ( α ) − σ )= X α ∈ F r ( σ ) f ( S ′ ( α )) g ( S ′ ( α ) − σ ) , and thus the convolution formula does not depend on S .By Proposition 2.8, any f ∈ A R ( G ; Σ) can be written in the form f = P U ∈F r U ˜1 U ,where F is a finite collection of compact open bisections of Σ. As the convolution productis clearly R -bilinear by definition, to show both that A R ( G ; Σ) is closed under convolutionand that the convolution product is associative, it suffices to show that ˜1 U ∗ ˜1 V = ˜1 UV for compact open bisections U and V of Σ (as the compact open bisections form a semi-group). Note that q | U and q | V are injective. The definition of the convolution shows thatsupp(˜1 U ∗ ˜1 V ) ⊆ ( R × · U )( R × · V ) = R × · U V . For ( τ, ρ ) ∈ ( U × V ) ∩ Σ (2) , choosinga section ζ : G → Σ such that ζ ( q ( τ )) = τ gives that if σ = τ ρ , then (˜1 U ∗ ˜1 V )( σ ) = P α ∈ G r ( σ ) ˜1 U ( ζ ( α ))˜1 V ( ζ ( α ) − σ ) = ˜1 U ( τ )˜1 V ( ρ ) = 1, and so ˜1 U ∗ ˜1 V agrees with ˜1 UV on U V .Thus ˜1 U ∗ ˜1 V = ˜1 UV by Lemma 2.7. (cid:3) Note that when the intended meaning is clear, we sometimes write f g rather than f ∗ g for the convolution product of two functions f, g ∈ A R ( G, σ ).The following corollary is a straightforward consequence of Proposition 2.8 and theproof of Proposition 2.9.
Corollary 2.10.
Let F and G be finite collections of compact open bisections of Σ withmutually disjoint images in G , and suppose that f, g ∈ A R ( G ; Σ) satisfy f = P U ∈F r U ˜1 U and g = P V ∈G t V ˜1 V . Then f ∗ g = X U ∈F ,V ∈G r U t V ˜1 UV . (2.2) In particular, each σ ∈ Σ is contained in a compact open bisection X of Σ , and (˜1 X ∗ f )( r ( σ ′ )) = f ( σ ′ ) = ( f ∗ ˜1 X )( s ( σ ′ )) for all σ ′ ∈ X. Proof.
Equation (2.2) follows from the formula ˜1 U ∗ ˜1 V = ˜1 UV established in the proofof Proposition 2.9, and from the bilinearity of multiplication. The second claim followseasily from equation (2.2). (cid:3) An important feature of the twisted Steinberg algebra A R ( G ; Σ) is that, even thoughit does not contain the algebra A R (Σ (0) ) of locally constant functions on the unit space ofΣ (because such functions are not R × -contravariant), it does contain an algebra canoni-cally isomorphic to A R (Σ (0) ), and the actions of elements of this algebra on A R ( G ; Σ) bymultiplication are exactly as one would expect. Proposition 2.11.
Let G be an ample Hausdorff groupoid and let (Σ , i, q ) be a discrete R -twist over G . The map f ˜ f obtained from Lemma 2.7 applied to elements f ∈ C c (Σ (0) , R ) ∼ = A R ( G (0) ) defines an injective R -algebra homomorphism of C c (Σ (0) , R ) into A R ( G ; Σ) whose image is precisely A R ( G (0) ; q − ( G (0) )) . For f ∈ C c (Σ (0) , R ) and a ∈ A R ( G ; Σ) , we have ( ˜ f ∗ a )( σ ) = f ( r ( σ )) a ( σ ) and ( a ∗ ˜ f )( σ ) = a ( σ ) f ( s ( σ )) for all σ ∈ Σ .Proof. That f ˜ f is an injective R -module map that has image contained in A R ( G (0) ; q − ( G (0) )) is straightforward. For the reverse containment, observe that if g ∈ A R ( G (0) ; q − ( G (0) )), then f := g | Σ (0) ∈ C c (Σ (0) , R ), and since ˜ f and g are elements of A R ( G (0) ; q − ( G (0) )) that agree on Σ (0) , they are equal by Lemma 2.7.That this map is multiplicative and hence is an R -algebra homomorphism is a particularcase of the final assertion (where a ∈ C c (Σ (0) , R )), so we now just have to prove the finalassertion. Fix f ∈ C c (Σ (0) , R ) and a ∈ A R ( G ; Σ). Let S : q (supp( f )) → supp( f ) be theidentity map. Then S is a section from supp G ( ˜ f ) to supp( ˜ f ), and so Proposition 2.9shows that for all σ ∈ Σ,( ˜ f ∗ a )( σ ) = X β ∈ G r ( σ ) ∩ supp G ( ˜ f ) ˜ f ( S ( β )) a ( S ( β ) − σ ) . The only term in the sum is β = r ( σ ), which yields S ( β ) = r ( σ ), and so the sum collapsesto ( ˜ f ∗ a )( σ ) = ˜ f ( r ( σ )) a ( σ ) = f ( r ( σ )) a ( σ ) since ˜ f extends f by definition.For the second equality, fix σ ∈ supp( a ) and choose a section S : supp G ( a ) → supp( a )such that S ( q ( σ )) = σ . Then( a ∗ ˜ f )( σ ) = X β ∈ G r ( σ ) ∩ supp G ( a ) a ( S ( β )) ˜ f ( S ( β ) − σ ) . The only nonzero terms in the sum are those corresponding to β ∈ G r ( σ ) ∩ supp G ( a ) suchthat S ( β ) − σ ∈ supp( ˜ f ) ⊆ i ( G (0) × R × ) . Since supp G ( ˜ f ) ∩ G r ( σ ) = { r ( σ ) } , the only possible nonzero summand is such that q ( S ( β ) − σ ) = r ( σ ); that is, β = q ( σ ). Since σ ∈ supp( a ), we have S ( β ) = σ by ourchoice of S , and so the sum collapses to ( a ∗ ˜ f )( σ ) = a ( σ ) ˜ f ( σ − σ ) = a ( σ ) f ( s ( σ )). If σ ∈ Σ \ supp( a ), then both ( a ∗ ˜ f )( σ ) and a ( σ ) f ( s ( σ )) are zero. (cid:3) ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 13
When R is indecomposable, the idempotents of A R ( G (0) ) are the characteristicfunctions of compact open subsets of G (0) . Given that indecomposability of R isimplied by condition (WT) when I ( B ) is nontrivial, the R -algebra homomorphism fromProposition 2.11 gives a better perspective on the elements of I ( B ). Corollary 2.12.
Let G be an ample Hausdorff groupoid, let R be an indecomposablecommutative ring, and let (Σ , i, q ) be a discrete R -twist over G . Then the map U ˜1 U gives a lattice isomorphism between compact open subsets of Σ (0) and I ( B ) . Normalisers and inverse semigroups.
An inverse semigroup is a semigroup S such that, for each s ∈ S , there is a unique element s † with ss † s = s and s † ss † = s † .We have ( st ) † = t † s † and ( s † ) † = s . The idempotents of an inverse semigroup form acommutative subsemigroup, and are precisely the elements of the form s † s . Details canbe found in [28]. Definition 2.13.
Let R be a commutative unital ring, let A be an R -algebra, and let B ⊆ A be a commutative R -subalgebra. Suppose that the set I ( B ) of idempotents of B is a set of local units for A : that is, for any { a , . . . , a n } ⊆ A , there exists e ∈ I ( B ) with ea i = a i = a i e for all i ∈ { , . . . , n } . We define the normaliser of B to be the set N ( B ) := { n ∈ A : there exists k ∈ A with knk = k, nkn = n, and kBn ∪ nBk ⊆ B } . We use the notation N A ( B ) for N ( B ) if we need to specify the ambient algebra A . Webegin by establishing that N ( B ) is an inverse semigroup. Lemma 2.14.
For each n ∈ N ( B ) , there exists a unique k ∈ N ( B ) such that knk = k, nkn = n, and kBn ∪ nBk ⊆ B. (2.3) Proof.
Fix n ∈ N ( B ). If k satisfies equation (2.3) for n , then k is itself in N ( B ) because n satisfies equation (2.3) for k . Suppose that k and k both satisfy equation (2.3) for n .We claim that since I ( B ) forms a set of local units for A , we have k n, k n, nk , nk ∈ B. Indeed, there exists e ∈ I ( B ) with k , k , n ∈ eAe , and so k i n = k i en ∈ B and nk i = nek i ∈ B for i ∈ { , } . Using this and that B is commutative for the third and sixthequalities, we obtain k = k nk = k nk nk = k nk nk = k nk = k nk nk = k nk nk = k nk = k . (cid:3) For each n ∈ N ( B ), we write n † for the unique element k ∈ N ( B ) satisfyingequation (2.3). So a ∈ A belongs to N ( B ) if and only if a † exists. Notice that the aboveproof shows that aa † , a † a ∈ I ( B ) for all a ∈ N ( B ). Also note that I ( B ) ⊆ N ( B ) since e † = e for each e ∈ I ( B ). Lemma 2.15.
The normaliser N ( B ) is an inverse semigroup with inverse n n † andset of idempotents I ( B ) .Proof. To see that N ( B ) is closed under multiplication, fix n, m ∈ N ( B ). Then nmBm † n † ⊆ nBn † ⊆ B , and similarly, m † n † Bnm ⊆ B . As observed above, mm † , n † n ∈ I ( B ), and so mm † and n † n commute. Hence( nm )( m † n † )( nm ) = nn † nmm † m = nm. Similarly, m † n † ( nm ) m † n † = m † n † , and so nm ∈ N ( B ). Thus N ( B ) is a semigroup.Lemma 2.14 therefore implies that N ( B ) is an inverse semigroup with inversion givenby n n † . The final statement follows because if e ∈ N ( B ) is an idempotent, then e = e † e ∈ I ( B ), as we saw earlier. (cid:3) Recall that there is a partial order on any inverse semigroup given by s ≤ t if and onlyif s = ss † t (or, equivalently, if and only if s = ts † s ). The partial order is preserved bymultiplication and inversion, and the product of two idempotents is their meet. Further-more, s ≤ t if and only if there is an idempotent e such that s = te , and this is equivalentto the existence of an idempotent f such that s = f t . See [28] for details. Note that ≤ reduces to the usual order relation on commuting idempotents; that is, e ≤ f if and onlyif ef = e . In Lemmas 2.16 and 2.17, we present various additional properties of ≤ thatwe use throughout the paper. The first of these results is straightforward, so we omit theproof. Lemma 2.16.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasuch that I ( B ) forms a set of local units for A . Fix t ∈ R × . (a) If n ∈ N ( B ) , then tn ∈ N ( B ) with ( tn ) † = t − n † . (b) If n, m ∈ N ( B ) with n ≤ m , then tn ≤ tm . Lemma 2.17.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasatisfying condition (WT) such that I ( B ) forms a set of local units for A . Then thefollowing hold. (a) If n ∈ N ( B ) and e ∈ I ( B ) satisfy n ≤ e , then n ∈ I ( B ) . (b) For f, e ∈ I ( B ) and t ∈ R × , if = f ≤ te , then t = 1 . (c) For n, m ∈ N ( B ) , we have nn † m = n ⇐⇒ mn † n = n . (d) For all n ∈ N ( B ) and e ∈ I ( B ) , we have en, ne ≤ n . (e) If e ∈ I ( B ) and n ∈ N ( B ) , then n − ne, n − en ∈ N ( B ) , with ( n − ne ) † = n † − en † and ( n − en ) † = n † − n † e , and n − ne, n − en ≤ n .Proof. Part (a) is [28, Proposition 1.4.7(5)], and parts (c) and (d) are contained in [28,Lemma 1.4.6]. For part (b), suppose that 0 = f ≤ te . Then tf e = f = 0 implies that f e = 0 and t f e = ( tf e ) = f = f = tf e. Multiplying both sides by t − then gives (1 − t ) f e = 0. Hence t = 1, as B satisfies (WT).Part (e) follows from (d) once we observe that n † ne ≤ n † n and hence n † n − n † ne ∈ I ( B ) ⊆ N ( B ). Thus n − ne = n ( n † n − n † ne ) ≤ n , and n − ne is an element of N ( B ) withinverse ( n † n − n † ne ) n † = n † − en † . A similar argument proves the claim about n − en . (cid:3) We define free normalisers by analogy (with suitable modifications) with the definitiongiven by Kumjian in the setting of C*-algebras.
Definition 2.18 (cf. [25, Def. 1]) . Let R be a commutative unital ring and let A be an R -algebra, with a commutative subalgebra B ⊆ A whose idempotents form a set of localunits for A . We say that n ∈ N ( B ) is a free normaliser if either n ∈ B or ( n † n )( nn † ) = 0(this latter condition is equivalent to n = 0, as n = n ( n † nnn † ) n ).3. Algebraic quasi-Cartan pairs
In this section we introduce our main objects of study—algebraic quasi-Cartan pairs.Our main theorem shows that these are precisely the pairs of algebras that arise fromdiscrete R -twists satisfying an appropriate local bisection hypothesis. We also definetwo additional types of pairs of algebras: algebraic diagonal pairs, and algebraic Cartanpairs. We prove later in this section that every diagonal pair is an algebraic Cartan pairand every algebraic Cartan pair is an algebraic quasi-Cartan pair. The diagonal pairscorrespond to twists over principal groupoids, while the Cartan pairs correspond to twistsover effective groupoids. ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 15
Before defining these pairs of algebras, we need to introduce an appropriate notion ofa conditional expectation from an algebra onto an abelian subalgebra.
Definition 3.1.
Let R be an indecomposable commutative ring, let A be an R -algebra,and let B ⊆ A be an abelian subalgebra such that I ( B ) is a set of local units for A . Amap P : A → B is called a conditional expectation if(i) P is R -linear;(ii) P | B = id B ; and(iii) P ( bab ′ ) = bP ( a ) b ′ for a ∈ A and b, b ′ ∈ B .The conditional expectation P : A → B is faithful if, for every a ∈ A \{ } , there exists n ∈ N ( B ) such that P ( na ) = 0.Note that since B contains local units for A , condition (iii) of Definition 3.1 impliesthat P is both left and right B -linear. Remark . Watatani [49] defines a conditional expectation P : A → B to be nondegen-erate if for every nonzero a ∈ A there exists a ′ ∈ A such that P ( a ′ a ) = 0. Since we willassume that A is spanned by N ( B ), the notion of a nondegenerate conditional expectationis equivalent to our definition of a faithful conditional expectation in our setting.We now define algebraic diagonal pairs, Cartan pairs, and quasi-Cartan pairs. Definition 3.3.
Let R be a unital ring, let A be an R -algebra, and let B ⊆ A be anabelian subalgebra satisfying condition (WT) and with the following properties.(i) The set I ( B ) forms a set of local units for A .(ii) B = span ( I ( B )).(iii) A = span ( N ( B )).(iv) There exists a faithful conditional expectation P : A → B .Then we say that the pair ( A, B ) is:(ADP) an algebraic diagonal pair if A is spanned by the free normalisers of B ;(ACP) an algebraic Cartan pair if B is a maximal abelian subalgebra of A ; and(AQP) an algebraic quasi-Cartan pair if there is a faithful conditional expectation P : A → B such that for each n ∈ N ( B ) there exists e ∈ I ( B ) such that P ( n ) = en = ne . Remarks . (1) To construct the inverse semigroup N ( B ) in Definition 2.13, we required B to beabelian and to satisfy property (i) of Definition 3.3. In fact, B being abelian andproperty (i) are almost enough to build our discrete twist Σ over G in the next twosections. However, we need property (AQP) of Definition 3.3 in Proposition 5.10to show that G is Hausdorff.(2) In the definition of an algebraic quasi-Cartan pair, note that if n ∈ N ( B ) satisfies P ( n ) = 0, then e = 0 satisfies P ( n ) = en = ne . So we could equivalently requirejust that whenever P ( n ) = 0 there exists e ∈ I ( B ) such that P ( n ) = en = ne .The simplest example of an algebraic diagonal pair is the following. Let R be anindecomposable commutative ring, let A = M n ( R ) be the ring of n × n matrices over R ,and let B be the subalgebra of diagonal matrices. Then B is spanned by the diagonalmatrix units E ii , which are idempotent, and A is spanned by the elementary matrix units E ij , which are free normalisers. The faithful conditional expectation is given by makingall non-diagonal entries zero. This example motivates the terminology.A more complicated, but illustrative, class of examples is the class of Leavitt pathalgebras associated to directed graphs. Recall that if E is a directed graph and L R ( E )is its Leavitt path algebra over a ring R , then it admits a commutative subalgebra D R ( E )—the subalgebra generated by the idempotents of the form s µ s µ ∗ for finitepaths µ . Example 9.5 shows that ( A R ( E ) , D R ( E )) is an algebraic quasi-Cartan pair.Proposition 7.1 together with known structure theory for the groupoid of a directedgraph show that ( A R ( E ) , D R ( E )) is an algebraic Cartan pair if and only if every cycle in E has an entrance, and is an algebraic diagonal pair if and only if E contains no cycles. Lemma 3.5.
Every algebraic diagonal pair is an algebraic Cartan pair.Proof.
Suppose that (
A, B ) is an algebraic diagonal pair and that a ∈ A commutes withevery element of B ; we must show that a ∈ B . Since ( A, B ) is an algebraic diagonal pair,we can express a as a finite linear combination of free normalisers. Every free normaliser n either belongs to B or satisfies n † nnn † = 0; so there exist b ∈ B , a finite set F ⊆ N ( B )such that n † nnn † = 0 for all n ∈ F , and coefficients α n ∈ R for each n ∈ F such that a = b + X n ∈ F α n n. Thus it suffices to show that a ′ := P n ∈ F α n n is zero. Since a and b are in the commutant(centraliser) of B , their difference a ′ = a − b is also in the commutant of B . Let M be theset of minimal nonzero idempotents in the Boolean algebra X generated by { n † n, nn † : n ∈ F } . Then each e ∈ M commutes with a ′ . Since P e ∈ M e is the maximum element of X , we have n P e ∈ M e = nn † n P e ∈ M e = nn † n = n . Hence a ′ = P e ∈ M a ′ e = P e ∈ M ea ′ e = P n ∈ F α n P e ∈ M ene . So it suffices to show that each ene = 0. Fix n ∈ F and e ∈ M . If n † ne = 0, then ene = enn † ne = 0, so we may suppose that n † ne = 0. Since e is a minimalnonzero idempotent in X , we deduce that en † n = e , and hence enn † = en † nnn † = 0, since n ∈ F . Thus ene = enn † ne = 0, as required. (cid:3) We next show that every algebraic Cartan pair is an algebraic quasi-Cartan pair, sothat we have (ADP) implies (ACP) implies (AQP).
Lemma 3.6.
Every algebraic Cartan pair is an algebraic quasi-Cartan pair. Moreover,every conditional expectation on an algebraic Cartan pair satisfies (AQP).Proof.
Suppose that (
A, B ) is an algebraic Cartan pair with conditional expectation P : A → B . To establish that ( A, B ) is an algebraic quasi-Cartan pair, suppose that n ∈ N ( B ) satisfies P ( n ) = 0. Write P ( n ) = P ki =1 t i e i , where the e i are mutually orthog-onal elements of I ( B ) and each t i ∈ R \{ } . Let e ( n ) := k X i =1 e i . Then e ( n ) ∈ I ( B ). We show that e ( n ) n = ne ( n ) = P ( n ). First, we claim that if f ≤ e ( n )with f ∈ I ( B ), then n † f n = f . To prove the claim, notice that P ( nf ) = P ( n ) f = k X i =1 t i e i f, and since nf n † ∈ nBn † ⊆ B , P ( nf ) = P ( nn † nf ) = P ( nf n † n ) = nf n † P ( n ) = P ( n ) nf n † = k X i =1 t i e i nf n † . (3.1)Since P ( nf ) f = P ( nf ), we deduce after multiplying both sides of equation (3.1) on theright by f that P ki =1 t i e i f = P ki =1 t i e i nf n † f . For any j ∈ { , . . . , k } , multiplying bothsides by e j on the left yields t j e j f = t j e j nf n † f , and so each t j ( e j f − e j nf n † f ) = 0.Since t j = 0 and e j f − e j nf n † f = e j f − e j f nf n † is an idempotent, we deduce from ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 17 condition (WT) that each e j f = e j nf n † f . Summing over all j ∈ { , . . . , k } and usingthat f ≤ e ( n ) yields f = e ( n ) f = k X i =1 e i f = k X i =1 e i nf n † f = e ( n ) nf n † f ≤ nf n † . Therefore, n † f n ≤ n † nf n † n ≤ f . To prove the reverse inequality, note that P ( nf ) = P ( n ) f = f P ( n ) = P ( f n ) = P ( nn † f n ) = P ( n ) n † f n = k X i =1 t i e i n † f n. Therefore, for any j ∈ { , . . . , k } , we have t j e j f = t j e j n † f n , forcing t j ( e j f − e j n † f n ) = 0 . Hence e j f = e j n † f n by (WT), as n † f n ≤ f implies that e j f − e j n † f n is an idempotent.Once again summing over j , and using that f ≤ e ( n ), yields f = e ( n ) n † f n ≤ n † f n , andso f = n † f n for all f ≤ e ( n ), proving the claim.It follows that if f ≤ e ( n ) in I ( B ), then nf = nn † f n = f nn † n = f n. (3.2)Now fix g ∈ I ( B ). Applying equation (3.2) to f = ge ( n ) ≤ e ( n ) at the first step, andto f = e ( n ) at the final step, we obtain g ( e ( n ) n ) = nge ( n ) = ne ( n ) g = ( e ( n ) n ) g. Since B is spanned by I ( B ), we deduce that e ( n ) n commutes with all elements of B .Since ( A, B ) is an algebraic Cartan pair, B is a maximal abelian subalgebra, and so e ( n ) n ∈ B . We then have e ( n ) n = P ( e ( n ) n ) = e ( n ) P ( n ) = P ( n ) by the definition of e ( n ). As we already observed, equation (3.2) gives e ( n ) n = ne ( n ), and so ( A, B ) is analgebraic quasi-Cartan pair. (cid:3)
In Section 6, we will use the expectation P : A → B satisfying condition (AQP) to con-struct an isomorphism of A onto the twisted Steinberg algebra of an associated twist. So a priori the isomorphism would depend upon the choice of conditional expectations satis-fying (AQP). We finish this section by showing that in fact such a conditional expectationis uniquely determined by the algebraic structure of ( A, B ). Proposition 3.7.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. For each nor-maliser n ∈ N ( B ) , the set { e ∈ I ( B ) : e ≤ n † n and ne ∈ B } contains a maximum element e ( n ) . If P is a conditional expectation satisfying (AQP), then P ( n ) = ne ( n ) = e ( n ) n . Inparticular, there is a unique conditional expectation P : A → B that satisfies (AQP).Proof. Fix n ∈ N ( B ) and suppose that P : A → B is a conditional expectation satisfy-ing (AQP). Write P ( n ) = P ki =1 t i e i , where the e i are mutually orthogonal idempotentsand each t i ∈ R \{ } . Let e ( n ) := P ki =1 e i ∈ I ( B ). Then we have P ( n ) e ( n ) = P ( n ). Weclaim that e ( n ) is the smallest such element of I ( B ). To see this, suppose that f ∈ I ( B )satisfies P ( n ) f = P ( n ). Then, for each j ∈ { , . . . , k } , we have t j f e j = P ( n ) f e j = P ( n ) e j = t j e j , and hence t j ( e j − f e j ) = 0, whence e j = f e j by (WT). Thus e ( n ) = f e ( n ) ≤ f , provingthe claim. Since P satisfies (AQP), there exists f ∈ I ( B ) such that f n = nf = P ( n ).Thus ne ( n ) = nf e ( n ) = P ( n ) e ( n ) = P ( n ) = e ( n ) P ( n ) = e ( n ) f n = f e ( n ) n = e ( n ) n. Notice that e ( n ) is an element of the set { e ∈ I ( B ) : e ≤ n † n and ne ∈ B } : we haveshown that ne ( n ) = P ( n ) ∈ B , and since P ( n )( n † n ) = P ( n ), we have e ( n ) ≤ n † n by ourprevious argument showing that e ( n ) is the smallest such element. It remains to showthat e ( n ) is a maximum. Fix g ∈ { e ∈ I ( B ) : e ≤ n † n and ne ∈ B } . We show that g ≤ e ( n ). Using that ng, g ∈ B , we compute ng = P ( ng ) = P ( n ) g = ne ( n ) g = nge ( n ),and so g = n † ng = n † nge ( n ) = ge ( n ) ≤ e ( n ). (cid:3) Corollary 3.8.
Suppose that ( A, B ) is an algebraic Cartan pair. Then there is a uniqueconditional expectation from A to B .Proof. Lemma 3.6 implies that every conditional expectation from A to B satisfies (AQP),and Proposition 3.7 implies that there is a unique such expectation. (cid:3) We conclude this section with an example showing that (AQP) is a generalisation ofthe much studied no-nontrivial-units condition on twisted group rings. Recall that if R is a commutative unital ring, G is a group with identity e , and c : G × G → R × isa normalised 2-cocycle, then the twisted group ring R ( G, c ) is the R -algebra of finitelysupported functions f : G → R with multiplication given by( f ∗ g )( β ) = X α ∈ G c ( α, α − β ) f ( α ) g ( α − β ) . It is a unital ring with identity the point-mass function δ e . Note that twisted group ringsare precisely the twisted Steinberg algebras of twists over discrete groups (viewed as amplegroupoids); see Lemma 4.3. Example . Let G be a discrete group with identity e , let R be an indecomposablecommutative ring, and let c : G × G → R × be a normalised 2-cocycle. Let A := R ( G, c )be the corresponding twisted group ring, and let B := Rδ e ∼ = R . Then I ( B ) = { , δ e } spans B (which trivially satisfies (WT)), and N ( B ) = A × ∪ { } , which clearly contains { δ g : g ∈ G } and hence spans A . We claim that ( A, B ) is an algebraic quasi-Cartan pairif and only if every unit of A is of the form tδ g for some t ∈ R × and g ∈ G ; that is, A only has trivial units.In this setup, a conditional expectation is an R -linear map P : A → Rδ e that fixes Rδ e . We claim that the only possible conditional expectation satisfying (AQP) is givenon the basis by fixing δ e and annihilating δ g if g = e . Indeed, δ g is a normaliser and hence P ( δ g ) = f δ g for some f ∈ I ( B ) = { , δ e } . But since P has image B , we must choose f = 0unless g = e , in which case we must choose f = δ e . This formula defines a conditionalexpectation P . It is faithful because if a ∈ A \{ } has a nonzero coefficient of δ g , then P ( δ g − a ) = 0.It remains to verify that P satisfies (AQP) if and only if A only has trivial units. If A has only trivial units and u ∈ N ( A ) = A × ∪ { } , then P ( u ) = u = uδ e = δ e u if u ∈ R × δ e , or P ( u ) = 0 = u u , otherwise, and so P satisfies (AQP). Conversely, if P satisfies (AQP) and u ∈ A × , then we can find g ∈ G with e in the support of uδ g − .Then P ( uδ g − ) = rδ e for some r = 0. On the other hand, P ( uδ g − ) = uδ g − f with f = 0or f = δ e . We conclude that rδ e = P ( uδ g − ) = uδ g − , and so rδ g = u ∈ A × and r ∈ R × .4. Building an algebraic quasi-Cartan pair from a twist
In this section we show that a twist gives rise to an algebraic quasi-Cartan pair if andonly if it satisfies an appropriate analogue of the local bisection hypothesis of [47]. We thenalso follow the arguments of [47] to see that an algebraic twist satisfies the local bisectionhypothesis if and only if the sub-twist corresponding to the interior of the isotropy doesso, and that a sufficient condition for this is that there is a dense set of units x for which ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 19 the twisted group algebra corresponding to the fibre of the interior of the isotropy over x has no nontrivial units. Finally, we demonstrate that if the underlying groupoid iseffective, then the twist gives rise to an algebraic Cartan pair, and if it is principal, thenwe obtain an algebraic diagonal pair. Definition 4.1.
Let R be a commutative unital ring. We say that a discrete R -twist(Σ , i, q ) over G satisfies the local bisection hypothesis if for every normaliser n of A R ( G (0) ; q − ( G (0) )) in A R ( G ; Σ), the set supp G ( n ) is a bisection of G .Our interest in twists satisfying the local bisection hypothesis is that they give riseto algebraic quasi-Cartan pairs. The following lemma can be seen as a generalisation ofExample 3.9. Lemma 4.2.
Let R be an indecomposable commutative ring and let (Σ , i, q ) be a discrete R -twist over G . Let A = A R ( G ; Σ) , and let B = A R ( G (0) ; q − ( G (0) )) ⊆ A . The pair ( A, B ) satisfies conditions (i)–(iv) of Definition 3.3 with faithful conditional expectation P : A → B given by restriction of functions from Σ to q − ( G (0) ) . If ( A, B ) is an algebraicquasi-Cartan pair, then P satisfies (AQP). The pair ( A, B ) is an algebraic quasi-Cartanpair if and only if (Σ , i, q ) satisfies the local bisection hypothesis.Proof. Since R is indecomposable, the idempotents of C c ( G (0) , R ) are precisely the charac-teristic functions of compact open subsets of G (0) , and hence B satisfies condition (WT),because B ∼ = C c ( G (0) , R ) by Proposition 2.11. A routine argument using Corollary 2.10shows that for each compact open bisection U of Σ, ˜1 U is a normaliser of B in A withinverse ˜1 † U = ˜1 U − .Properties (i) and (ii) of Definition 3.3 follow immediately from Proposition 2.11, andproperty (iii) follows from Proposition 2.8. For (iv), it is straightforward to see that P is a conditional expectation, and for faithfulness, note that if f ( σ ) = 0 for some σ ∈ Σ,then for any compact open bisection X ⊆ Σ containing σ , Corollary 2.10 implies that P (˜1 X f )( r ( σ )) = (˜1 X f )( r ( σ )) = f ( σ ) = 0.Suppose that ( A, B ) is an algebraic quasi-Cartan pair. Then by Proposition 3.7, thereis a unique faithful conditional expectation P ′ : A → B satisfying (AQP). We claim that P = P ′ . Since the functions ˜1 V with V ⊆ Σ a compact open bisection span A asan R -module by Proposition 2.8, it suffices by the linearity of P and P ′ to prove thatfor each such V , P ′ (˜1 V ) = ˜1 V | q − ( G (0) ) = ˜1 W , where W := V ∩ q − ( G (0) ), which is acompact open bisection contained in V . Recall from Corollary 2.12 that U ˜1 U is alattice isomorphism between compact open subsets of Σ (0) and I ( B ). By Proposition 3.7(and several applications of Corollary 2.10), there is a maximum element ˜1 U of I ( B )satisfying ˜1 U ≤ ˜1 † V ˜1 V = ˜1 s ( V ) such that ˜1 V ˜1 U ∈ B , and P ′ (˜1 V ) = ˜1 V ˜1 U = ˜1 V U . Itfollows that
V U ⊆ W by the definition of B (and since U ⊆ Σ (0) ). On the other hand, s ( W ) ⊆ s ( V ) and ˜1 V ˜1 s ( W ) = ˜1 W ∈ B , and so it follows by maximality that s ( W ) ⊆ U .Thus W = V s ( W ) ⊆ V U , and so W = V U . We conclude that P ′ (˜1 V ) = ˜1 W = P (˜1 V ), asrequired.Now, we show that (Σ , i, q ) satisfies the local bisection hypothesis. Suppose that n isa normaliser and that σ ∈ supp( n ). We show that if τ ∈ Σ s ( σ ) \ ( R × · σ ), then n ( τ ) = 0;that the same holds if τ ∈ Σ r ( σ ) \ ( R × · σ ) follows by a similar argument. By choosinga continuous local section from G to Σ, we can find a compact open bisection U ⊆ Σcontaining σ such that ( r, ǫ ) r · ǫ is a homeomorphism of R × × U onto q − ( q ( U )). Thenwe have P ( n ˜1 U − ) = ( n ˜1 U − ) | q − ( G (0) ) , and in particular, by Corollary 2.10, P ( n ˜1 U − )( r ( σ )) = n ( σ )˜1 U − ( σ − ) = n ( σ ) = 0 . Since P ( n ˜1 U − ) = n ˜1 U − ˜1 V for some compact open subset V ⊆ G (0) , and we deduce that r ( σ ) ∈ V . Using again that P ( n ˜1 U − ) = ( n ˜1 U − ) | q − ( G (0) ) , we see that n ( τ ) = ( n ˜1 U − )( τ σ − ) = ( n ˜1 U − ˜1 V )( τ σ − ) = P ( n ˜1 U − )( τ σ − ) . Since τ ∈ Σ s ( σ ) \ ( R × · σ ), we have τ σ − / ∈ q − ( G (0) ), so we deduce that n ( τ ) = 0.Finally, suppose that (Σ , i, q ) satisfies the local bisection hypothesis. We must showthat ( A, B ) is an algebraic quasi-Cartan pair. Fix a normaliser n ∈ N ( B ). Then supp G ( n )is a bisection. Let U := supp( n ) ∩ Σ (0) , and let e := ˜1 U ∈ I ( B ). Then ne = n | s − ( U ) . If n ( σ ) = 0 and σ / ∈ R × · U = supp( n ) ∩ q − ( G (0) ), then s ( σ ) / ∈ U because supp G ( n ) is abisection. It follows by a routine calculation that ne = n | s − ( U ) = P ( n ), and similarly en = P ( n ), as required. (cid:3) Since twists satisfying the local bisection hypothesis give examples of algebraic quasi-Cartan pairs, we are interested in identifying conditions that guarantee the local bisectionhypothesis. In the next few results, we follow the analysis used in [47] to give a sufficientcondition in terms of twisted group algebras associated to the fibres of the interior of theisotropy in the twist. We begin by describing these algebras. This is standard, and ourproof follows ideas from [4, Section 4.3].Given a discrete R -twist (Σ , i, q ) over a groupoid G , let I denote the interior of theisotropy in G , and let J := q − ( I ), which is the interior of the isotropy in Σ. Notice thatthe image of i is contained in J and we can restrict q to J and obtain a map q | J : J → I .It is straightforward to check that ( J , i, q | J ) is again a twist and that A R ( I ; J ) can beseen as a subalgebra of A R ( G ; Σ) by extending an element of A R ( I ; J ) to be zero outside J . Similarly, if we fix x ∈ G (0) , then I x = I x = I xx , and J x = J x = J xx , and we canbuild a discrete R -twist ( J x , i x , q x ) over I x , where i x = i | { x }× R × and q x = q | J x . Lemma 4.3.
Let R be a commutative unital ring and let (Σ , i, q ) be a discrete R -twistover G . Let I denote the interior of the isotropy in G and let J := q − ( I ) , which is theinterior of the isotropy in Σ . Fix x ∈ G (0) , and let ζ : I x → J x be a section for q | J x suchthat ζ ( x ) = x . Define c x : I x × I x → R × by ζ ( α ) ζ ( β ) = c x ( α, β ) · ζ ( αβ ) . Then c x is anormalised -cocycle on I x , and there is an isomorphism ρ ζ : A R ( I x ; J x ) → R ( I x , c x ) of R -algebras such that ρ ζ ( f )( β ) = f ( ζ ( β )) for all f ∈ A R ( I x ; J x ) and β ∈ I x .Proof. It is routine to verify that each c x is a normalised 2-cocycle using associativity ofmultiplication in Σ and that ζ preserves the identity. Certainly there is an R -linear map ρ ζ satisfying the given formula. To see that ρ ζ is injective, suppose that ρ ζ ( f ) = 0. Thenfor each σ ∈ J x , we have σ = t · ζ ( q ( σ )) for some t ∈ R × , and then f ( σ ) = f ( t · ζ ( q ( σ ))) = t − f ( ζ ( q ( σ ))) = t − ρ ζ ( f )( q ( σ )) = 0. To see that ρ ζ is surjective, fix h ∈ R ( I x , c x ). Foreach σ ∈ J x , there is a unique t σ ∈ R × such that σ = t σ · ζ ( q ( σ )) and t σ = 1 for σ ∈ ζ ( I x ).Define f : J x → R by f ( σ ) = t − σ h ( q ( σ )) for all σ ∈ J x . Then f ∈ A R ( I x ; J x ), and ρ ζ ( f ) = h .It remains to check that ρ ζ is multiplicative. Fix f, g ∈ A R ( I x ; J x ) and β ∈ I x . Then ρ ζ ( f ∗ g )( β ) = ( f ∗ g )( ζ ( β )) . By the definition of convolution in A R ( I x ; J x ), the convolution product ( f ∗ g )( ζ ( β )) canbe computed by choosing any section from I x to J x , so we obtain ρ ζ ( f ∗ g )( β ) = X α ∈I x f ( ζ ( α )) g ( ζ ( α ) − ζ ( β )) . For each α ∈ I x , we have ζ ( α ) ζ ( α − ) = c x ( α, α − ) ζ ( αα − ) = c x ( α, α − ) · x , and rear-ranging this shows that ζ ( α ) − = c x ( α, α − ) − ζ ( α − ). Therefore, since ζ ( α − ) ζ ( β ) = ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 21 c x ( α − , β ) · ζ ( α − β ) for all α ∈ I x , we have ρ ζ ( f ∗ g )( β ) = X α ∈I x c x ( α, α − ) f ( ζ ( α )) g ( ζ ( α − ) ζ ( β ))= X α ∈I x c x ( α, α − ) c x ( α − , β ) − f ( ζ ( α )) g ( ζ ( α − β )) . The 2-cocycle identity and that c x is normalised give c x ( α − , β ) c x ( α, α − β ) = c x ( α, α − ) c x ( αα − , β ) = c x ( α, α − ) . Thus, rearranging gives c x ( α, α − ) c x ( α − , β ) − = c x ( α, α − β ). Hence ρ ζ ( f ∗ g )( β ) = X α ∈I x c x ( α, α − β ) f ( ζ ( α )) g ( ζ ( α − β ))= X α ∈I x c x ( α, α − β ) ρ ζ ( f )( α ) ρ ζ ( g )( α − β )= ( ρ ζ ( f ) ∗ ρ ζ ( g ))( β ) . (cid:3) Changing the section ζ in Lemma 4.3 results in a cohomologous normalised 2-cocycleand hence it does not change the algebra (up to isomorphism), nor does it affect whetheror not the twisted group ring has only trivial units. In fact, a twisted group ring isgraded by the underlying group, and the property of having only trivial units meansthat all the units are in the homogeneous component of the identity; the isomorphism ofalgebras corresponding to replacing a 2-cocycle by a cohomologous one does not changethe grading. Remark . Let R be an indecomposable commutative ring and let (Σ , i, q ) be a discrete R -twist over G . The argument of [47, Proposition 4.5] shows that if n is a normaliser of A R ( G (0) ; q − ( G (0) )) in A R ( G ; Σ), then n † n ∈ A R ( G (0) ; q − ( G (0) )) is equal to ˜1 s (supp( n )) and nn † = ˜1 r (supp( n )) . Proposition 4.5.
Let R be an indecomposable commutative ring and let (Σ , i, q ) be adiscrete R -twist over G . Let I denote the interior of the isotropy in G and let J := q − ( I ) ,which is the interior of the isotropy in Σ . Let x ∈ G (0) , let I x and c x be as in Lemma 4.3,and suppose that the twisted group ring R ( I x , c x ) has only trivial units. Then for anynormaliser n of A R ( G (0) ; q − ( G (0) )) in A R ( I ; J ) , we have | supp I ( n ) ∩ I x | ≤ .Proof. Suppose that n | J x = 0. We must show that n | J x is supported on q − ( µ ) for some µ ∈ I . Fix α ∈ J x with n ( α ) = 0. Remark 4.4 gives n † n = 1 s (supp( n )) = 1 r (supp( n )) = nn † . So we obtain ( n † n )( s ( α )) = 1 = ( nn † )( s ( α )). Fix a section ζ : I x → J x andlet ρ ζ : A R ( I x ; J x ) → R ( I x , c x ) be the isomorphism of Lemma 4.3. Since J = q − ( I )consists of isotropy, the convolution product n † n evaluated at x is the same as n † | J x ∗ n | J x at x (and similarly for nn † ), and we obtain n † | J x ∗ n | J x = 1 A R ( I x ; J x ) = n | J x ∗ n † | J x . Itfollows that ρ ζ ( n | J x ) is a unit, and therefore it has the form tδ µ for some µ ∈ I x . So n | J x = tρ − ζ ( δ µ ) is supported on R × · ζ ( µ ) = q − ( µ ), as required. (cid:3) The following lemma can be viewed as a far-reaching extension of Example 3.9.
Lemma 4.6.
Let R be an indecomposable commutative ring and let (Σ , i, q ) be a discrete R -twist over G . Let I denote the interior of the isotropy in G and J := q − ( I ) , which isthe interior of the isotropy in Σ . For each x ∈ G (0) , let I x and c x be as in Lemma 4.3.Suppose that the set of units x ∈ G (0) for which R ( I x , c x ) × = { tδ α : t ∈ R × and α ∈ I x } is dense in G (0) . Then ( J , i, q | J ) satisfies the local bisection hypothesis. Proof.
Suppose that n is a normaliser of A R ( G (0) ; q − ( G (0) )) in A R ( I ; J ). We must showthat supp I ( n ) is a bisection. Suppose for contradiction that this is not the case. Thenwe can find α, β ∈ supp I ( n ) with s ( α ) = s ( β ) but α = β . Since I is Hausdorff andsupp I ( n ) is open, we can find disjoint compact open bisections U and V of I with α ∈ U , β ∈ V , and U, V ⊆ supp I ( n ). By assumption, s ( U ) ∩ s ( V ) = ∅ as s ( α ) = s ( β ) is in theintersection. Since s ( U ) ∩ s ( V ) is an open subset of G (0) , there exists x ∈ s ( U ) ∩ s ( V )such that the twisted group ring R ( I x , c x ) has only trivial units. Take α ′ ∈ U and β ′ ∈ V with s ( α ′ ) = x = s ( β ′ ). Then α ′ = β ′ by Proposition 4.5, contradicting that U and V aredisjoint. (cid:3) The next lemma shows that the local bisection hypothesis can be checked on the interiorof the isotropy.
Lemma 4.7.
Let R be an indecomposable commutative ring and let (Σ , i, q ) be a discrete R -twist over G . Let I denote the interior of the isotropy in G and let J := q − ( I ) ,which is the interior of the isotropy in Σ . Let A := A R ( G ; Σ) , let A ′ := A R ( I ; J ) , and let B := A R ( G (0) ; q − ( G (0) )) . Then (a) the normaliser N A ′ ( B ) of B in A ′ satisfies N A ′ ( B ) = N A ( B ) ∩ A ′ ; and (b) (Σ , i, q ) satisfies the local bisection hypothesis if and only if ( J , i, q | J ) does.Proof. The proof very closely follows the arguments of [47, Proposition 4.7, Corollary 4.8,and Proposition 4.10], with just minor adjustments to incorporate the generality of twists.Claim 1 (cf. [47, Proposition 4.7]): if n ∈ N A ( B ) and σ, τ ∈ supp( n ), then s ( σ ) = s ( τ )if and only if r ( σ ) = r ( τ ). We suppose that x = s ( σ ) = s ( τ ) and prove that r ( σ ) = r ( τ ); the converse follows from a symmetric argument. Suppose for contradiction that y := r ( σ ) = r ( τ ) =: z . Fix disjoint compact open neighbourhoods y ∈ U and z ∈ V inΣ (0) , and let p := ˜1 U and q := ˜1 V . Then pn, qn ∈ N A ( B ).Remark 4.4 gives ( n † pn )( x ) = 1 because ( pn )( σ ) = 0 and ( pn ) † ( pn ) = n † pn . Similarly,( n † qn )( x ) = 1 because ( qn )( τ ) = 0. Since p + q = ˜1 U ∪ V ∈ I ( B ) by the disjointness of U and V , we have by the same reasoning that ( n † ( p + q ) n )( x ) = 1 because (( p + q ) n )( σ ) = 0.Hence 2 = ( n † pn )( x ) + ( n † qn )( x ) = ( n † ( p + q ) n )( x ) = 1 , which is a contradiction.Claim 2 (cf. [47, Corollary 4.8]): if n ∈ N A ( B ), then supp( n ) supp( n ) − andsupp( n ) − supp( n ) are contained in J . For this, suppose that σ ∈ supp( n ) and τ ∈ supp( n ) − and s ( σ ) = r ( τ ). Then τ − ∈ supp( n ) and s ( σ ) = s ( τ − ), so Claim 1shows that r ( σ ) = r ( τ − ). So στ is in the isotropy of Σ. It follows that the open setsupp( n ) supp( n ) − is contained in the isotropy, and hence supp( n ) supp( n ) − ⊆ J . Asymmetric argument shows that supp( n ) − supp( n ) ⊆ J .Now for part (a), note that N A ′ ( B ) is clearly contained in N A ( B ) ∩ A ′ . For the reversecontainment, it suffices to show that if n ∈ N A ( B ) ∩ A ′ , then supp( n † ) ⊆ J . So supposethat σ ∈ supp( n † ). Then Remark 4.4 shows that ( nn † )( s ( σ )) = 1. So any section ζ : G s ( σ ) → Σ s ( σ ) satisfies1 = ( nn † )( s ( σ )) = X α ∈ G s ( σ ) n ( ζ ( α ) − ) n † ( ζ ( α )) . Since the sum is nonzero, at least one term, say corresponding to α ∈ G s ( σ ) , satisfies ζ ( α ) − ∈ supp( n ) and ζ ( α ) ∈ supp( n † ). Applying Claim 1 to n † , we see that r ( α ) = r ( σ ). Since supp( n ) ⊆ J by hypothesis on n , we have α − ∈ I , and so we deduce that s ( σ ) = s ( α ) = r ( α ) = r ( σ ), as required. ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 23
For part (b), the “only if” implication is clear, so we suppose that ( J , i, q | J ) satisfies thelocal bisection hypothesis, and establish that (Σ , i, q ) does too. Fix n ∈ N A ( B ). Supposethat σ, τ ∈ supp( n ) satisfy s ( σ ) = s ( τ ). Then r ( σ ) = r ( τ ) by Claim 1. We must show that q ( σ ) = q ( τ ). Since n is locally constant, there is a compact open bisection U containing σ such that n | U is constant. Claim 2 implies that U − supp( n ) ⊆ J . Let m := ˜1 U − . Then mn belongs to N A ( B ) ∩ A ′ , and so part (a) shows that mn ∈ N A ′ ( B ). Since ( J , i, q | J )satisfies the local bisection hypothesis, we deduce that supp G ( mn ) = supp I ( mn ) is abisection. Let x = s ( σ ). We claim that x, σ − τ ∈ supp( mn ). To see this, fix a section S : supp G ( m ) = q ( U − ) → supp( m ) = R × · U − with S ( q ( σ )) = σ . Since U is a bisection,we have ( mn )( x ) = X γ ∈ G x m ( S ( γ )) n ( S ( γ ) − ) = n ( S ( q ( σ ))) = n ( σ ) = 0 , and ( mn )( σ − τ ) = X γ ∈ G x m ( S ( γ )) n ( S ( γ ) − σ − τ ) = n ( τ ) = 0 , as claimed. Since q (supp( mn )) is a bisection and s ( x ) = x = s ( σ − τ ), we deduce that q ( σ − τ ) = q ( x ), and hence q ( σ ) = q ( τ ). (cid:3) We have seen that twists satisfying the local bisection hypothesis give rise to algebraicquasi-Cartan pairs. To finish the section we show that if the underlying groupoid G iseffective, then we obtain an algebraic Cartan pair, and if it is principal, then we obtainan algebraic diagonal pair. Proposition 4.8.
Let R be an indecomposable commutative ring and let (Σ , i, q ) be adiscrete R -twist over G . If G is effective then ( A, B ) := ( A R ( G ; Σ) , A R ( G (0) ; q − ( G (0) ))) is an algebraic Cartan pair, and if G is principal then ( A, B ) is an algebraic diagonal pair.Proof. If G is effective, then Lemma 4.7(b) implies that the local bisection hypothesisholds. Then by Lemma 4.2, ( A, B ) is an algebraic quasi-Cartan pair with conditionalexpectation P : A → B given by restriction of functions from Σ to q − ( G (0) ). So, we justhave to show that if G is effective, then ( A, B ) satisfies (ACP) and that if G is principalthen ( A, B ) satisfies (ADP).First suppose that G is effective. Then for f ∈ A \ B , we must show that f does notcommute with B . Since f / ∈ B , there exists σ ∈ Σ \ q − ( G (0) ) such that f ( σ ) = 0.Then U := f − ( f ( σ )) is open. Since G is effective, q ( U ) is open, and q ( σ ) / ∈ G (0) , [9,Lemma 3.1(3)] implies that there exists τ ∈ U such that r ( τ ) = s ( τ ). Fix a compactopen set V ⊆ G (0) that contains s ( τ ) but not r ( τ ), and let e V := ˜1 V ∈ B . Then( f e V )( τ ) = f ( τ ) = f ( σ ) = 0 = ( e V f )( τ ). So f e V = e V f , and hence f does not commutewith B , as required.Now suppose that G is principal. Fix a compact open bisection U ⊆ Σ. ByProposition 2.8, it suffices to show that ˜1 U is a linear combination of free normalisers.Let U := U ∩ q − ( G (0) ) and V := U \ U . Both U and V are compact open bisections,and we have ˜1 U = ˜1 U + ˜1 V . Since ˜1 U ∈ B is a free normaliser, we just have toshow that ˜1 V is a linear combination of free normalisers. Since q ( V ) ∩ G (0) is emptyand G is principal, we have r ( σ ) = s ( σ ) for all σ ∈ V . Thus, for each σ ∈ V , wecan find a compact open bisection V σ ⊆ V containing σ with V σ = ∅ by choosingdisjoint compact open neighbourhoods U, U ′ ⊆ G (0) of r ( σ ) and s ( σ ), respectively, andputting V σ := U V U ′ ⊆ V . By compactness, we can cover V by finitely many compactopen bisections V , . . . , V n such that V i = ∅ for i ∈ { , . . . , n } . Then by putting V ′ i = V i \ S i − j =1 V ′ j for each i ∈ { , . . . , n } , we obtain a refined cover of V consisting ofmutually disjoint sets. Now each ˜1 V ′ i is a free normaliser, and ˜1 V = P ni =1 ˜1 V ′ i . (cid:3) Building a twist from a pair of algebras
Throughout this section we assume that A is an R -algebra with B an abelian subalgebrawithout torsion in the sense of condition (WT) (that is, for e ∈ I ( B ) and t ∈ R , if te = 0,then t = 0 or e = 0), and satisfying property (i) of Definition 3.3 (that is, I ( B ) is a set oflocal units for A ).5.1. The groupoid Σ.
In this section we use the partial order ≤ on N ( B ) defined inSection 2.5 (on page 14) to define the groupoid of ultrafilters on N ( B ), which will playthe same role as Renault’s Weyl twist in [43].If U ⊆ N ( B ), then the upclosure of U is the set U ↑ := { m ∈ N ( B ) : there exists n ∈ U with n ≤ m } . A filter of N ( B ) is a subset U ⊆ N ( B ) \{ } such that U = U ↑ and whenever m, n ∈ U there exists p ∈ U such that p ≤ m, n . An ultrafilter is a maximal filter. Thecollection Σ of all ultrafilters of N ( B ) forms a groupoid with the following structure(see [28, Proposition 9.2.1] and [29, Proposition 2.13]). For each U ∈ Σ, U − := { n † : n ∈ U } . A pair (
U, V ) of ultrafilters of N ( B ) is composable if and only if m † mnn † = 0 (or,equivalently, if and only if mn = 0) for all m ∈ U and n ∈ V , and then U V := { mn : m ∈ U, n ∈ V } ↑ . The definition of an ultrafilter ensures that a pair (
U, V ) of ultrafilters is composable ifand only if s ( U ) := U − U is equal to r ( V ) := V V − . It follows thatΣ (0) = { U ∈ Σ : U ∩ I ( B ) = ∅ } . For n ∈ N ( B ), we write V n := { U ∈ Σ : n ∈ U } . The collection {V n : n ∈ N ( B ) } forms a basis of open bisections for a topology on Σ,making it an ´etale groupoid. We have V n V m = V nm and V − n = V n † for all m, n ∈ N ( B ).For more details, see [5, Lemma 3.2]. By [30, Lemma 2.22] (see also [5, Propositions 2.2and 4.4]), Σ (0) is a Hausdorff subspace of Σ. In Lemma 5.3, we show how scalar multipli-cation in the algebra interacts with the groupoid structure. Remark . Note that [5, Proposition 2.2 and Proposition 4.4] say that U U ∩ I ( B )is a homeomorphism from the set of ultrafilters of N ( B ) that contain an element of I ( B )to the set of ultrafilters of I ( B ), and so we often identify elements of Σ (0) with ultrafiltersof I ( B ) when convenient. In particular, since I ( B ) is a Boolean algebra, Stone dualityimplies that V e is compact for each e ∈ I ( B ).We will frequently use the following standard fact about ultrafilters. This fact, andmuch of what we say about the groupoid Σ as an ample groupoid, would immediatelyfollow from the results of [31] if one proved that N ( B ) is a Boolean inverse semigroup(called a weakly Boolean inverse semigroup in [31]). However, to prove this would go toofar afield. Lemma 5.2.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasatisfying condition (WT) and (i) of Definition 3.3. Let Σ be the groupoid of ultrafiltersof N ( B ) . Suppose that n ∈ U ∈ Σ and that e , . . . , e k ∈ I ( B ) are mutually orthogonalidempotents satisfying n = P ki =1 ne i . Then there exists a unique i ∈ { , . . . , k } such that ne i ∈ U . Dually, if n = P ki =1 f i n for mutually orthogonal idempotents f , . . . , f k ∈ I ( B ) ,then f i n ∈ U for a unique i ∈ { , . . . , k } . ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 25
Proof.
Without loss of generality, we may assume that n † ne i = e i for each i ∈ { , . . . , k } ,and hence n † n = P ki =1 e i . Since n † n ∈ U − U ∩ I ( B ), which is an ultrafilter of theBoolean algebra I ( B ) by Remark 5.1, it follows that there is a unique i ∈ { , . . . , k } with e i ∈ U − U ∩ I ( B ) (since the characteristic function of an ultrafilter on a Boolean algebrais a Boolean algebra homomorphism). Then ne i ∈ U ( U − U ) = U . Moreover, if ne j ∈ U for some j ∈ { , . . . , k } , then e j = ( ne j ) † ne j ∈ U − U ∩ I ( B ), and so j = i . This provesthe first statement.For the dual result, note that n = P ki =1 f i n = P ki =1 n ( n † f i n ), and that the n † f i n aremutually orthogonal idempotents in I ( B ). Therefore, f i n = n ( n † f i n ) ∈ U for a unique i ∈ { , . . . , k } by the previous paragraph. (cid:3) Lemma 5.3.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasatisfying condition (WT) and satisfying property (i) of Definition 3.3. Let Σ be thegroupoid of ultrafilters of N ( B ) . For t, s ∈ R × and n ∈ N ( B ) , we have the following. (a) If U ∈ Σ , then tU := { tm : m ∈ U } ∈ Σ . (b) If U ∈ Σ , then ( tU ) − = t − U − . (c) If ( U, W ) ∈ Σ (2) , then ( tU, sW ) ∈ Σ (2) and ( tU )( sW ) = ( ts )( U W ) . (d) If U ∈ Σ , then s ( U ) = s ( tU ) and r ( U ) = r ( tU ) . (e) t V n := { tU : U ∈ V n } = V tn . (f) If t = 1 , then V tn ∩ V n = ∅ . (g) If U, W ∈ Σ (0) with tU = sW , then t = s and U = W .Proof. For part (a), notice that tU ⊆ N ( B ) by Lemma 2.16(a). We first show that tU is afilter. Fix tm, tn ∈ tU . Since U is a filter, there exists k ∈ U such that k ≤ m and k ≤ n .Then tk ∈ tU , and Lemma 2.16(b) implies that tk ≤ tm and tk ≤ tn . Next suppose that p ∈ N ( B ) and that there exists tm ∈ tU with tm ≤ p . Then m ≤ t − p by Lemma 2.16(b)again. Again using that U is a filter, we have t − p ∈ U , and hence p = tt − p ∈ tU . Also,0 / ∈ tU , for otherwise 0 = t − ∈ U . Hence tU is a filter.To see that tU is an ultrafilter, suppose that tU ⊆ W , where W is a filter. Then U ⊆ t − W . Since U is an ultrafilter and t − W is a filter by our previous argument, U = t − W , and hence tU = W .For part (b), Lemma 2.16(a) implies that( tU ) − = { ( tm ) † : m ∈ U } = { t − m † : m ∈ U } = t − U − . For part (c), fix (
U, W ) ∈ Σ (2) . We begin by verifying that tU and sW are composable.Suppose that m ∈ tU and n ∈ sW . Then m = tm ′ with m ′ ∈ U and n = sn ′ with n ′ ∈ W . Since ( U, W ) ∈ Σ (2) , we have that m ′ n ′ = 0. Therefore, mn = ( ts ) m ′ n ′ = 0 since ts ∈ R × . Thus ( tU, sW ) ∈ Σ (2) . Since ( ts )( U W ) is an ultrafilter by part (a), it sufficesto show that ( ts )( U W ) ⊆ ( tU )( sW ). Fix k ∈ ( ts )( U W ). Then there exists m ∈ U W such that k = ( ts ) m . Therefore, there exist n ∈ U and p ∈ W such that np ≤ m , and so( tn )( sp ) = ( ts )( np ) ≤ ( ts ) m = k . Hence k ∈ ( tU )( sW ), and so ( tU )( sW ) = ( ts )( U W ).Item (d) follows easily from parts (b) and (c).For part (e), the containment t V n ⊆ V tn follows from part (a). For the reverse con-tainment, fix W ∈ V tn . Then tn ∈ W ∈ Σ, and hence n ∈ t − W ∈ Σ by part (a). So t − W ∈ V n , and W = t ( t − W ) ∈ t V n .For part (f), we prove the contrapositive. For this, suppose that U ∈ V tn ∩ V n . Then n = 0 and tn, n ∈ U , and so there exists m ∈ U such that 0 = m ≤ tn, n . Thus0 = mm † ≤ tnn † , and so t = 1 by Lemma 2.17(b).For part (g), suppose that U, W ∈ Σ (0) satisfy tU = sW . Then U = ( t − s ) W , and soit suffices to show that t − s = 1. For this, fix e , e ∈ I ( B ) with e ∈ U and e ∈ W . Then t − se ∈ U , and so there exists f ∈ U such that 0 = f ≤ e , t − se . Now byLemma 2.17(a), we have f ∈ I ( B ), and so Lemma 2.17(b) implies that t − s = 1. (cid:3) The groupoid G . In this section we assume that A is an R -algebra, and that B isan abelian subalgebra satisfying condition (WT) and satisfying properties (i) and (ii) ofDefinition 3.3.We begin by defining a relation ≃ on Σ, the groupoid of ultrafilters of the inversesemigroup N ( B ), by U ≃ W ⇐⇒ there exists t ∈ R × such that U = tW. (5.1)Then ≃ is an equivalence relation. We define G to be the quotient Σ / ≃ endowed withthe quotient topology, and we denote the corresponding quotient map by q : Σ → G . Inthe following lemma we show that G is a groupoid that inherits its structure from Σ. Lemma 5.4.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasatisfying condition (WT) and satisfying properties (i) and (ii) of Definition 3.3. Let Σ be the groupoid of ultrafilters of N ( B ) , and let q : Σ → G be the quotient map definedabove. (a) The quotient G = { q ( U ) : U ∈ Σ } is a groupoid with inversion given by q ( U ) − := q ( U − ) , composable pairs G (2) := { ( q ( U ) , q ( W )) : ( U, W ) ∈ Σ (2) } , and compositiongiven by q ( U ) q ( W ) := q ( U W ) for all ( U, W ) ∈ Σ (2) . We have s ( q ( U )) = q ( s ( U )) and r ( q ( U )) = q ( r ( U )) for all U ∈ Σ , and so G (0) = q (Σ (0) ) . (b) The quotient map q : Σ → G is a groupoid homomorphism. (c) The quotient map restricts to a bijection q | Σ (0) : Σ (0) → G (0) .Proof. For part (a), note that Lemma 5.3(b) implies that the inverse is well-defined. Com-posability and the product are well-defined by Lemma 5.3(c). The remainder of thegroupoid structure comes from the groupoid structure of Σ, and part (b) follows as well.For part (c), first note that q (Σ (0) ) = G (0) by part (a). Injectivity of q | Σ (0) follows fromLemma 5.3(g) with t = 1. (cid:3) Lemma 5.5.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasatisfying condition (WT) and satisfying properties (i) and (ii) of Definition 3.3. Let Σ be the groupoid of ultrafilters of N ( B ) , and let q : Σ → G be the quotient map definedabove. (a) The collection {V n : n ∈ N ( B ) } forms a basis of compact open bisections for thetopology on Σ . In particular, Σ is an ample groupoid. (b) The quotient map is open and restricts to a homeomorphism of unit spaces. (c)
The collection { q ( V n ) : n ∈ N ( B ) } forms a basis of compact open bisections forthe quotient topology on G . In particular, G is an ample groupoid.Proof. For part (a), to see that Σ is ample, we show that V n is compact for each n ∈ N ( B ).Fix n ∈ N ( B ). Then V n is homeomorphic to r ( V n ) because V n is a bisection. Now r ( V n ) = V n V n † = V nn † , which is compact by Remark 5.1.For part (b), since G is endowed with the quotient topology by definition, to see that q is open, it suffices to show that q − ( q ( V n )) is open in Σ for any n ∈ N ( B ). UsingLemma 5.3(e) for the last equality, we calculate q − ( q ( V n )) = { tU : t ∈ R × and U ∈ V n } = [ t ∈ R × t V n = [ t ∈ R × V tn , ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 27 which is open in Σ. That q restricts to a homeomorphism of unit spaces follows fromLemma 5.4(c).For part (c), compactness of each q ( V n ) follows from part (a), because q is continuous.Since {V n : n ∈ N ( B ) } is a basis for Σ and q is surjective, the collection { q ( V n ) : n ∈ N ( B ) } covers G . Part (b) shows that each q ( V n ) is open in G . So to see that { q ( V n ) : n ∈ N ( B ) } is a basis for the quotient topology, it suffices to show that for eachopen subset O of G and each element q ( U ) ∈ O , there exists n ∈ N ( B ) such that q ( U ) ∈ q ( V n ) ⊆ O. Since O is open in the quotient topology on G , q − ( O ) is open in Σ. Since U ∈ q − ( O ) and {V n : n ∈ N ( B ) } is a basis for Σ, there exists n ∈ N ( B ) such that q ( U ) ∈ q ( V n ) ⊆ O , andso { q ( V n ) : n ∈ N ( B ) } is a basis. To see that the sets q ( V n ) are open bisections, we needto show that the source and range maps are injective on each q ( V n ). But this follows fromparts (a) and (c) of Lemma 5.4, since V n is an open bisection. Now [8, Proposition 6.6]implies that G is a topological groupoid, and hence G is ample. (cid:3) The twist of an algebraic quasi-Cartan pair.
The main theorem of this sectionis that if (
A, B ) is an algebraic quasi-Cartan pair, then (Σ , i, q ) is a discrete R -twist. Theorem 5.6.
Suppose that A is an R -algebra, and that B ⊆ A is an abelian subalgebrasatisfying condition (WT) and satisfying properties (i) and (ii) of Definition 3.3. Let Σ bethe groupoid of ultrafilters of N ( B ) , let G be the quotient of Σ by the equivalence relationgiven in equation (5.1) , and let q : Σ → G be the quotient map. Define i : G (0) × R × → Σ by i ( q ( U ) , t ) = tU for U ∈ Σ (0) and t ∈ R × . Then the sequence G (0) × R × i ֒ → Σ q ։ G satisfies all of the properties of a discrete R -twist except, perhaps, that G is Hausdorff. If ( A, B ) is an algebraic quasi-Cartan pair, then the sequence is a discrete R -twist. Since B is without torsion as in condition (WT), [24, Th´eor`eme 1] shows that there isan isomorphism φ : B → A R (Σ (0) ) that satisfies φ ( e ) = 1 V e for all e ∈ I ( B ) . (5.2)To start the proof of Theorem 5.6, we show that if ( A, B ) is an algebraic quasi-Cartanpair then G is Hausdorff (see Proposition 5.10). For this, we need the following lemma. Lemma 5.7.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. If n ∈ N ( B ) ∩ B ,then n † ∈ B , and φ ( n )( U ) ∈ R × ∪ { } for all U ∈ Σ (0) .Proof. Suppose that n ∈ N ( B ) ∩ B . Then n † = n † nn † = P ( n † n ) n † = P ( n † ) nn † = P ( n † nn † ) = P ( n † ) ∈ B, proving the first statement. Since n † n is an idempotent element of B and is a rightinverse for n , and since R is indecomposable, we have φ ( n † n ) = 1 W for some compactopen set W containing the support of φ ( n ). Since φ is multiplicative, we deduce that φ ( n † )( U ) φ ( n )( U ) = 1 for all U ∈ supp( φ ( n )), and so each φ ( n † )( U ) is an inverse for φ ( n )( U ) (because R is commutative). (cid:3) The following consequence of Lemma 5.7 is used frequently, often without comment.
Corollary 5.8.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. Let n ∈ N ( B ) ∩ B ,and suppose that n = P ki =1 t i e i , where e , . . . , e k are mutually orthogonal idempotents and t , . . . , t k ∈ R \{ } . Then t i ∈ R × for each i ∈ { , . . . , k } . Proof.
Choose an ultrafilter W ∈ V e i . Then e j ∈ W if and only if j = i , and so φ ( n )( W ) = t i ∈ R × by Lemma 5.7. (cid:3) Remark . If (
A, B ) is an algebraic quasi-Cartan pair, then since B = span ( I ( B )),every n ∈ B can be expressed in the form n = P ki =1 t i e i , where e , . . . , e k ∈ I ( B ) aremutually orthogonal idempotents and t , . . . , t k ∈ R . It follows (using Corollary 5.8) that n ∈ N ( B ) ∩ B if and only if t , . . . , t k ∈ R × ∪ { } . Thus N ( B ) ∩ B = (cid:8) b ∈ B : φ ( b ) ∈ R × ∪ { } (cid:9) . Proposition 5.10.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. Then G isHausdorff.Proof. We know that G is ample by Lemma 5.5(c), and an ample groupoid is Hausdorff ifand only if its unit space is closed. So we prove that G (0) is closed. Since q is a quotientmap, it suffices to show that q − ( G \ G (0) ) is open. This set consists of all ultrafilters U of N ( B ) such that U contains no elements of the form te with e ∈ I ( B ) and t ∈ R × .Let U be such an ultrafilter, and fix n ∈ U . By (AQP), there exists f ∈ I ( B ) such that P ( n ) = nf = f n ∈ N ( B ) ∩ B . Take mutually orthogonal idempotents e , . . . , e k ∈ I ( B )and coefficients t , . . . , t k ∈ R \{ } such that nf = P ( n ) = P ki =1 t i e i , and note that t i ∈ R × for each i ∈ { , . . . , k } , by Corollary 5.8. Also, nf e i = t i e i for all i ∈ { , . . . , k } ,and so nf = P ki =1 nf e i .Note that n − nf ∈ N ( B ) by Lemma 2.17(e). Thus, since n = n ( n † n − n † nf ) + nf , wededuce that either nf ∈ U or n ( n † n − n † nf ) = n − nf ∈ U by Lemma 5.2. But if nf ∈ U ,then since nf = P ki =1 nf e i , we must have t i e i = nf e i ∈ U for some i ∈ { , . . . , k } by Lemma 5.2, which is a contradiction to our hypothesis on U . Thus we must have n − nf ∈ U , and so U ∈ V n − nf .We claim that if V ∈ V n − nf , then V / ∈ q − ( G (0) ). To see this, suppose for contradictionthat tg ∈ V for some t ∈ R × and g ∈ I ( B ). Then tg and n − nf have a commonlower bound m in V , and so t − m ≤ g . Hence t − m ∈ I ( B ) by Lemma 2.17(a), and so m = th for some idempotent h ∈ I ( B ). Since th = m ≤ n − nf ≤ n , we have that th = tht − hn = hn . So th = P ( th ) = P ( hn ) = hP ( n ) = hnf = thf. But then m = th = thf ≤ ( n − nf ) f = 0, and so m = 0, which contradicts m ∈ V .Thus V / ∈ q − ( G (0) ), as required. We conclude that q − ( G \ G (0) ) is open and hence G (0) is closed, whence G is Hausdorff. (cid:3) Proof of Theorem 5.6.
Lemma 5.4(b) shows that q is a groupoid homomorphism. It iscontinuous and surjective by definition. By Lemma 5.5(b), q restricts to a homeomor-phism of unit spaces. In what follows, we identify G (0) with Σ (0) for cleaner notation.To see that i is a groupoid homomorphism, fix a composable pair (( U, t ) , ( U, s )) ∈ ( G (0) × R × ) (2) , where U ∈ Σ (0) . Then i (( U, t )( U, s )) = i ( U, ts ) = tsU = tsU U. An application of Lemma 5.3(c) shows that this is equal to tU sU = i ( U, t ) i ( U, s ), andhence i is a groupoid homomorphism.To see that i is continuous, fix a basic open set V n ⊆ Σ. Then i − ( V n ) = { ( U, t ) ∈ Σ (0) × R × : tU ∈ V n } = { ( U, t ) ∈ Σ (0) × R × : U ∈ V t − n } by Lemma 5.3(e)= [ t ∈ R × ( V t − n ∩ Σ (0) ) × { t } . ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 29
Since V t − n ∩ Σ (0) is open in Σ (0) ∼ = G (0) , we deduce that i − ( V n ) is open in G (0) × R × .Lemma 5.3(g) shows that i is injective, and clearly i ( G (0) × { } ) = Σ (0) . For theexactness condition (DT1), fix U ∈ G (0) , and note that i ( { U } × R × ) = { tU : t ∈ R × } = q − ( U ) . We next check condition (DT3), which requires that the image of i is central in Σ. Fix U ∈ Σ and t ∈ R × . Then we have i ( r ( U ) , t ) U = t ( U U − ) U = tU = U ( t ( U − U )) = U i ( s ( U ) , t )by Lemma 5.3(c).We use the implication (4) = ⇒ (1) of Proposition 2.2 to complete the proof that thesequence satisfies (DT1)–(DT3) (and hence is a discrete R -twist when G is Hausdorff).We have that Σ is ample and that the map q is open by parts (a) and (b) of Lemma 5.5.We show that i is open. Fix a basic open set V e × { t } with e ∈ I ( B ) and t ∈ R × . Then,using Lemma 5.3(e) for the last equality, we see that i ( V e × { t } ) = t V e = V te . Thus i is an open map, as required.The final statement now follows by Proposition 5.10. (cid:3) The isomorphism A ∼ = A R ( G ; Σ)Let ( A, B ) be an algebraic quasi-Cartan pair and let Σ be the groupoid of ultrafilters of N ( B ). Let C (Σ , R ) denote the R -module of continuous (or equivalently, locally constant)functions from Σ to R with pointwise operations. In this section we build a map from A to C (Σ , R ) using both the faithful conditional expectation P : A → B satisfying (AQP)and the isomorphism φ : B → A R (Σ (0) ) from equation (5.2) that satisfies φ ( e ) = 1 V e forall e ∈ I ( B ), and we prove that this map is in fact an isomorphism of A onto the twistedSteinberg algebra A R ( G ; Σ). Proposition 6.1.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair, and let G and Σ be the groupoids constructed in Section 5. Let φ : B → A R (Σ (0) ) be the isomorphismfrom equation (5.2) that satisfies φ ( e ) = 1 V e for all e ∈ I ( B ) . For each a ∈ A , there is afunction b a : Σ → R such that for any ultrafilter U ∈ Σ and any n ∈ U , b a ( U ) = φ ( P ( n † a ))( s ( U )) . Furthermore, (a) b a is continuous; (b) the map a b a from A to C (Σ , R ) is R -linear; (c) the map a b a from A to C (Σ , R ) is injective; (d) b a ( tU ) = t − b a ( U ) for every t ∈ R × and U ∈ Σ ; (e) for b ∈ B , we have b b | Σ (0) = φ ( b ) , and supp( b b ) ⊆ i ( G (0) × R × ) .Proof. To see that there exists a (well-defined) function b a satisfying the given formula,we must show that φ ( P ( n † a ))( s ( U )) is independent of the choice of n ∈ U . So fix a ∈ A , U ∈ Σ, and n, m ∈ U . We must show that φ ( P ( n † a ))( s ( U )) − φ ( P ( m † a ))( s ( U )) = 0 . Observe that if m, n ∈ U , then there exists k ∈ U such that k ≤ m, n , whence k † ≤ m † , n † .So k † km † = k † = k † kn † , and moreover, φ ( k † k )( s ( U )) = 1 V k † k ( U − U ) = 1 . Now, using that φ is multiplicative and that P is a conditional expectation, we compute φ ( P ( n † a ))( s ( U )) − φ ( P ( m † a ))( s ( U )) = φ ( k † k )( s ( U )) φ ( P ( n † a − m † a ))( s ( U ))= φ ( P ( k † kn † a − k † km † a ))( s ( U ))= φ ( P ( k † a − k † a ))( s ( U ))= 0 . For part (a), fix a ∈ A . We show that b a is locally constant. Let U ∈ Σ and choose n ∈ U . The function φ ( P ( n † a )) ∈ A R (Σ (0) ) is locally constant, and so there is anidempotent e ∈ s ( U ) such that for each ultrafilter W ∈ Σ (0) containing e , we have φ ( P ( n † a ))( W ) = Φ( P ( n † a ))( s ( U )). Let m ∈ U with m † m ≤ e . Then m and n have acommon lower bound k ∈ U , and hence k † k ≤ m † m ≤ e . Thus k = nk † k ≤ ne . So ne ∈ U , and hence V ne is an open neighbourhood of U . If V ∈ V ne , then n ∈ V , because ne ≤ n (by Lemma 2.17(d)), and e ∈ s ( V ) because ( ne ) † ne = en † ne ≤ e . Therefore, b a ( V ) = φ ( P ( n † a ))( s ( V )) = φ ( P ( n † a ))( s ( U )) = b a ( U ) , by the choice of e . Thus b a is locally constant, and hence is continuous.Part (b) follows from the R -linearity of P and φ .For part (c), suppose that b a = 0. Then b a ( U ) = 0 for all U ∈ Σ. We claim that φ ( P ( na )) = 0 for all n ∈ N ( B ). Fix n ∈ N ( B ) and U ∈ Σ (0) . We show that φ ( P ( na ))( U ) = 0 by considering two cases. First, suppose that nn † ∈ U . Then U ∈ V nn † = V n V n † = s ( V n † ) , and so we can find an ultrafilter W ∈ V n † such that s ( W ) = U . We then have φ ( P ( na ))( U ) = φ ( P ( na ))( s ( W )) = b a ( W ) = 0 . For the second case, suppose that nn † / ∈ U . Then U / ∈ V nn † , and since P is a conditionalexpectation, we have φ ( P ( na ))( U ) = φ ( P ( nn † na ))( U ) = φ ( nn † )( U ) φ ( P ( na ))( U ) = 1 V nn † ( U ) φ ( P ( na ))( U ) = 0 , as claimed. Since φ is injective, we deduce that P ( na ) = 0 for all n ∈ N ( B ). Now, since P is faithful (property (iv) of Definition 3.3), we deduce that a = 0.For part (d), fix U ∈ Σ and t ∈ R × . Then s ( U ) = s ( tU ) by Lemma 5.3(d). Let n ∈ U .Then tn ∈ tU and ( tn ) † = t − n † by Lemma 2.16(a). Therefore, b a ( tU ) = φ ( P (( tn ) † a ))( s ( tU )) = φ ( P ( t − n † a ))( s ( U )) = t − φ ( P ( n † a ))( s ( U )) = t − b a ( U ) , as required.For part (e), fix b ∈ B and U ∈ Σ (0) with e ∈ U ∩ I ( B ), and note that e = e † . Then b b ( U ) = φ ( P ( eb ))( s ( U )) = φ ( eb )( U ) = φ ( e )( U ) φ ( b )( U ) = 1 V e ( U ) φ ( b )( U ) = φ ( b )( U ) . Finally, we show that the support of b b is contained in i ( G (0) × R × ). Since B is spannedby I ( B ) and the map a b a is R -linear by part (b), it suffices to consider the case where b = e ∈ I ( B ). Fix U ∈ Σ such that b e ( U ) = 0. We claim that U ∈ R × · Σ (0) . To see this,fix n ∈ U . Then φ ( P ( n † e ))( s ( U )) = b e ( U ) = 0 . Since (
A, B ) is an algebraic quasi-Cartan pair, there exists f ∈ I ( B ) such that f n † = n † f = P ( n † ), and so f en † = ef n † = eP ( n † ) = P ( n † ) e = n † f e. ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 31
Also, P ( n † e ) = P ( n † ) e = n † f e . We then have0 = φ ( P ( n † e ))( s ( U )) = φ ( n † f e )( s ( U ))= φ ( n † f ef e )( s ( U ))= φ ( n † f e )( s ( U )) φ ( f e )( s ( U ))= φ ( n † f e )( s ( U )) 1 V fe ( s ( U )) . Thus f e ∈ s ( U ), and so nf e ∈ U s ( U ) = U . Therefore, f en † ∈ U − . Since P ( n † e ) = n † f e ∈ N ( B ) ∩ B , Remark 5.9 implies that f en † = n † f e = P ( n † e ) can be expressed asa finite sum P ki =1 t i e i , where e , . . . , e k ∈ I ( B ) are mutually orthogonal idempotents and t , . . . , t k ∈ R × . Notice that for each i ∈ { , . . . , k } , we have t i e i = f en † e i , and so k X i =1 f en † e i = f en † ∈ U − . Since U − is an ultrafilter, it follows from Lemma 5.2 that there exists (a unique) i ∈{ , . . . , k } such that t i e i = f en † e i ∈ U − . Hence t − i e i = ( t i e i ) † ∈ U by Lemma 2.16(a),which forces V := t i U ∈ Σ (0) . Therefore, U = t − i V ∈ q − ( G (0) ) = i ( G (0) × R × ). (cid:3) We now prove that if (
A, B ) is an algebraic quasi-Cartan pair, and G and Σ are asconstructed in Section 5, then there is an isomorphism of A onto A R ( G ; Σ) that carries B to the canonical subalgebra isomorphic to A R (Σ (0) ). We need the following technicallemma.In what follows, we write b P ( a ) := [ P ( a ) for each a ∈ A , where a b a : A → C (Σ , R ) isthe map from Proposition 6.1. Lemma 6.2.
Let R be an indecomposable commutative ring. Suppose that ( A, B ) is analgebraic quasi-Cartan pair with faithful conditional expectation P : A → B satisfying(AQP). Let G and Σ be the groupoids constructed in Section 5. Let m, n ∈ N ( B ) and U ∈ Σ (0) . If b P ( mn )( U ) = 0 , then there exists f ∈ U ∩ I ( B ) such that f mn = mnf = b P ( mn )( U ) f ∈ B .Proof. Suppose that b P ( mn )( U ) = 0. Then P ( mn ) = 0 by Proposition 6.1(b), and by(AQP), there exists e ∈ I ( B ) such that e mn = mne = P ( mn ). Since P ( mn ) ∈ B ,Proposition 6.1(e) implies that0 = b P ( mn )( U ) = φ ( P ( mn ))( U ) = φ ( mm † mne )( U )= φ ( mm † )( U ) φ ( mne )( U ) φ ( e )( U ) = 1 V mm † ( U ) φ ( mne )( U ) 1 V e ( U ) , and so mm † , e ∈ U . Hence e := e mm † ∈ U ∩ I ( B ) satisfies emn = ee mn = eP ( mn ) = P ( mn ) e = mne e = mne. (6.1)Since eP ( mn ) ∈ B , there is a finite set F ⊆ I ( B ) of mutually orthogonal idempotents andnonzero coefficients r f ∈ R such that eP ( mn ) = P f ∈ F r f f ; without loss of generality wemay assume that f ≤ e for each f ∈ F . Thus, using Proposition 6.1(e) and equation (5.2),we see that 0 = b P ( mn )( U ) = b e ( U ) b P ( mn )( U ) = X f ∈ F r f b f ( U ) = X f ∈ F r f V f ( U ) . Hence we deduce that U ∈ S f ∈ F V f . But the mutual orthogonality of the idempotents in F implies that the V f are pairwise disjoint, and so there is a unique f ∈ F ∩ U . Therefore, r f = b P ( mn )( U ). Since f ≤ e , a similar argument to the one used in equation (6.1) showsthat f commutes with mn , and f mn = f emn = f eP ( mn ) = r f f = b P ( mn )( U ) f ∈ B . (cid:3) Lemma 6.3.
Let ( A, B ) be an algebraic quasi-Cartan pair, and let G and Σ be thegroupoids constructed in Section 5. Then for n ∈ N ( B ) and U ∈ Σ , we have b n ( U ) = ( t − if U ∈ V tn for some t ∈ R × otherwise . In particular, b n is equal to the function ˜1 V n ∈ A R ( G ; Σ) of Lemma 2.7.Proof. For the first case, suppose that U ∈ V tn for some t ∈ R × , and let φ : B → A R (Σ (0) )be the isomorphism of equation (5.2) that satisfies φ ( e ) = 1 V e for all e ∈ I ( B ). Then( tn ) † = t − n † by Lemma 2.16(a), and so n † n = ( tn ) † tn ∈ s ( U ). Therefore, we have b n ( U ) = φ ( P ( t − n † n ))( s ( U )) = φ ( t − n † n )( s ( U )) = t − φ ( n † n )( s ( U )) = t − . For the second case, we prove the contrapositive. Suppose that b n ( U ) = 0, and fix m ∈ U . Then by Proposition 6.1(e) and the definition of b n , we have b P ( m † n )( s ( U )) = φ ( P ( m † n ))( s ( U )) = b n ( U ) = 0 . Hence Lemma 6.2 shows that there exists an idempotent f ∈ s ( U ) such that f m † n = m † nf = b n ( U ) f . Since b n ( U ) f = f m † n ∈ N ( B ) ∩ B , Lemma 5.7 implies that t := b n ( U )belongs to R × . Since f ∈ s ( U ) and m ∈ U , we have mf ∈ U . Note that t ( mf ) = m ( tf ) = mm † nf ≤ n by Lemma 2.17(d). Thus mf ≤ t − n by Lemma 2.16(b), and so t − n ∈ U .That is, U ∈ V t − n , as required. (cid:3) For the surjectivity in the main theorem (Theorem 6.6), we need to know that eachelement of A R ( G ; Σ) can be written as a finite sum of elements of the form t b n (where t ∈ R and n ∈ N ( B )); we do this in the following two results, the first of which is standard. Lemma 6.4.
Let H be an ample groupoid and let D be a compact open bisection of H .Let B be an inverse semigroup of compact open bisections that form a basis for H andwhose idempotents are closed under relative complement and disjoint union. Then D canbe expressed as a finite disjoint union of elements of B .Proof. Since B is a basis and D is compact, we can certainly write D = S Ni =1 B i with B i ∈B . Put B ′ = B . Assume inductively that we have found B ′ , . . . , B ′ j ∈ B pairwise disjointwith S ji =1 B i = F ji =1 B ′ i for 1 ≤ j < n . Then put B ′ j +1 = B j +1 (cid:0) s ( B j +1 ) \ ∪ ji =1 s ( B ′ i ) (cid:1) . Byassumption on B , we have that B ′ j +1 ∈ B as B ′ i , B ′ k ⊆ D disjoint implies that s ( B ′ i ) and s ( B ′ k ) are disjoint because D is a bisection. Note that B ′ j +1 is disjoint from B ′ , . . . , B ′ j byconstruction.Since B ′ j +1 ⊆ B j +1 , by the inductive assumption it suffices to show that B j +1 ⊆ F j +1 i =1 B ′ i .Fix γ ∈ B j +1 . If s ( γ ) / ∈ s ( B ′ ) ∪ · · · ∪ s ( B ′ j ), then trivially γ ∈ B ′ j +1 . If s ( γ ) ∈ s ( B ′ i ) forsome i ∈ { , . . . , j } , then since B ′ i , B j +1 ⊆ D and D is a bisection, we must have that γ ∈ B ′ i . This completes the proof. (cid:3) Proposition 6.5.
Let ( A, B ) be an algebraic quasi-Cartan pair, and let G and Σ be thegroupoids constructed in Section 5. For any f ∈ A R ( G ; Σ) , there exist n , . . . , n M ∈ N ( B ) and t , . . . , t M ∈ R such that f = M X j =1 t j b n j . Proof.
By Proposition 2.8, it suffices to prove the claim for f = ˜1 D , where D is a compactopen bisection of Σ. The collection B = {V n : n ∈ N ( B ) } is an inverse semigroup ofcompact open bisections forming a basis for the topology on Σ. Moreover, if e, f ∈ I ( B ),then V e \ V f = V e − ef , and V e ∩ V f = ∅ if and only if ef = 0, in which case, V e ⊔ V f = V e + f , ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 33 by Stone duality applied to I ( B ). Thus, by Lemma 6.4, we can express D a finite disjointunion of basic compact open sets, say, D = F Mj =1 V n j . Then g := P Mj =1 b n j is an elementof A R ( G ; Σ) satisfying supp( g ) ⊆ R × · D and g | D ≡
1, by Lemma 5.3(e) and Lemma 6.3.Therefore, Lemma 2.7 gives g = ˜1 D . (cid:3) Theorem 6.6.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. Let G and Σ be thegroupoids constructed in Section 5. Then the map ϕ : a b a from A to C (Σ , R ) defined inProposition 6.1 is an isomorphism of A onto A R ( G ; Σ) that takes B to A R ( G (0) ; q − ( G (0) )) ,which is isomorphic to A R (Σ (0) ) , and hence also to A R ( G (0) ) .Proof. We begin by showing that b a ∈ A R ( G ; Σ) for each a ∈ A . If n ∈ N ( B ), then b n ∈ A R ( G ; Σ) by Lemma 6.3. Since each a ∈ A can be expressed as an R -linear combinationof elements of N ( B ) (by property (iii) of Definition 3.3), it follows that b a ∈ A R ( G ; Σ),because ϕ is R -linear by Proposition 6.1(b).Proposition 6.1 implies that ϕ is an injective R -linear map, and Proposition 6.5 impliesthat ϕ is surjective.To complete the proof that ϕ is an isomorphism, we must show that c aa ′ = b a ∗ b a ′ for all a, a ′ ∈ A . Since A is the R -linear span of N ( B ), it suffices to prove this for a = n, a ′ = m ∈ N ( B ). But this follows from Lemma 6.3 and Corollary 2.10, since V n V m = V nm .Finally, we check the statement about B . If e ∈ I ( B ), then ϕ ( e ) = b e = ˜1 V e byLemma 6.3. Since I ( B ) spans B and { ˜1 V e : e ∈ I ( B ) } spans A R ( G (0) ; q − ( G (0) )) (byProposition 2.11), it follows that ϕ carries B isomorphically to A R ( G (0) ; q − ( G (0) )), com-pleting the proof. (cid:3) Algebraic information from the isotropy structure of G In this section we describe the properties of the groupoid G (from the previous section)that identify the algebraic diagonal pairs and the algebraic Cartan pairs amongst allalgebraic quasi-Cartan pairs. Proposition 7.1.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. Then (a) ( A, B ) is an algebraic Cartan pair if and only if G is effective, and (b) ( A, B ) is an algebraic diagonal pair if and only if G is principal. The proof will follow easily once we establish the following two lemmas.
Lemma 7.2.
Let ( A, B ) be an algebraic Cartan pair, and let n ∈ N ( B ) . Suppose that V n ⊆ Iso(Σ) . Then n ∈ B .Proof. Let P : A → B be the faithful conditional expectation satisfying (AQP), and let φ : B → A R (Σ (0) ) be the isomorphism from equation (5.2) that satisfies φ ( e ) = 1 V e forall e ∈ I ( B ). We claim that nb = bn for every b ∈ B . Since B is spanned by I ( B )(property (ii) of Definition 3.3), it suffices to fix e ∈ I ( B ) and show that en = ne . Weshow that c ne = c en ; the claim then follows by the injectivity of a b a (Proposition 6.1(c)).Fix U ∈ Σ. If b n ( U ) = 0, then by multiplicativity of a b a (Theorem 6.6), we have c en ( U ) = b e ( U ) b n ( U ) = 0 = b n ( U ) b e ( U ) = c ne ( U ) . So it suffices to consider U ∈ supp( b n ) = R × · V n by Lemma 6.3; so, in particular, r ( U ) = s ( U ) since V n ∈ Iso(Σ). Now fix m ∈ U . Then c ne ( U ) = φ ( P ( m † ne ))( s ( U ))= φ ( P ( m † n ))( s ( U )) φ ( e )( s ( U ))= ( φ ( P ( m † n ))( s ( U )) if e ∈ s ( U )0 otherwise= (b n ( U ) if e ∈ s ( U )0 otherwise . In the first case, we have e ∈ s ( U ) = r ( U ). Since r ( U ) U = U , we have em ∈ U , andhence c en ( U ) = φ ( P ( m † en ))( s ( U )) = φ ( P (( em ) † n ))( s ( U )) = b n ( U ) = c ne ( U ) . For the second case, suppose that e / ∈ s ( U ) = r ( U ). Let e := mm † − emm † and let e := emm † . These are orthogonal idempotents in I ( B ), and m = e m + e m . Therefore,either e m = m − em or e m = em belongs to U , by Lemma 5.2. But if em ∈ U , thensince em ( me ) † = emm † e ≤ e , we have e ∈ r ( U ), which is a contradiction. So we musthave m − em ∈ U . Let k := ( m − em ) † . Then k = m † − m † e by Lemma 2.17(e), and so ke = 0. Therefore, c en ( U ) = φ ( P ( ken ))( s ( U )) = 0, as required. This completes the proofof the claim.By the claim, n commutes with all of B . Since B is maximal abelian, this forces n ∈ B . (cid:3) Lemma 7.3.
Suppose that ( A, B ) is an algebraic quasi-Cartan pair. If n ∈ N ( B ) ∩ B ,then V n ⊆ q − ( G (0) ) .Proof. The result is trivial if n = 0, and so we assume that n = 0. By Remark 5.9, thereexist mutually orthogonal idempotents e , . . . , e k ∈ I ( B ) and coefficients t , . . . , t k ∈ R × such that n = P ki =1 t i e i . Note that ne i = te i for each i ∈ { , . . . , k } , and so n = P ki =1 ne i .Fix V ∈ V n . Then Lemma 5.2 implies that ne i ∈ V for some (unique) i ∈ { , . . . , k } .Thus t i e i = ne i ∈ V , and so e i ∈ t − i V . Therefore, t − i V ∈ Σ (0) , and so V = t i ( t − i V ) ∈ R × · Σ (0) = q − ( G (0) ), as required. (cid:3) Proof of Proposition 7.1.
For part (a), first suppose that (
A, B ) is an algebraic Cartanpair. Fix an open set O contained in the interior of the isotropy of G . Without loss ofgenerality, we may assume that O = q ( V n ) for some n ∈ N ( B ). Then V n is contained inthe isotropy of Σ. Lemma 7.2 implies that n ∈ B , and Lemma 7.3 gives q ( V n ) ⊆ G (0) .Now suppose that G is effective. Then the algebraic quasi-Cartan pair ( A, B ) is isomor-phic to ( A R ( G ; Σ) , A R ( G (0) , q − ( G (0) ))) (by Theorem 6.6), which is an algebraic Cartanpair by Proposition 4.8.For part (b), first suppose that ( A, B ) is an algebraic diagonal pair. Fix U ∈ Σ such that r ( U ) = s ( U ). We must show that q ( U ) ∈ G (0) . For this, fix n ∈ U . Since A is spannedby the free normalisers of B , we can write n = n + P ki =1 n i , where n ∈ B and each n i is a normaliser satisfying n i = 0. Since s ( U ) is an ultrafilter, we can find an idempotent e ∈ s ( U ) = r ( U ) such that for every i ∈ { , . . . , k } m we have e ≤ n † i n i whenever n † i n i ∈ s ( U ), and en † i n i = 0 whenever n † i n i / ∈ s ( U ). Since e ∈ s ( U ) = r ( U ), we have P ki =0 en i e = ene ∈ U . If i ∈ { , . . . , k } satisfies n † i n i / ∈ s ( U ), then en i e = en i n † i n i e = 0;and if i ∈ { , . . . , k } satisfies n † i n i ∈ s ( U ), then e ≤ n † i n i , and so en i e = e ( n † i n i )( n i e ) = 0 ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 35 as well. So we obtain ene = en e ∈ U ∩ B ∩ N ( B ). Hence U ∈ V ene ⊆ q − ( G (0) ) byLemma 7.3.Now suppose that G is principal. Then ( A, B ) ∼ = ( A R ( G ; Σ) , A R ( G (0) , q − ( G (0) ))) is thealgebra of a twist over a principal groupoid (by Theorem 6.6), and hence is an algebraicdiagonal pair by Proposition 4.8. (cid:3) Recovering a twist from its quasi-Cartan pair
In this section we show that if (Σ , i, q ) is a discrete R -twist over a groupoid G , then wecan construct a natural embedding of Σ into the groupoid of ultrafilters Σ ′ obtained fromTheorem 5.6 applied to the pair ( A R ( G ; Σ) , A R ( G (0) ; q − ( G (0) ))). We then show that thismap is an isomorphism if and only if (Σ , i, q ) satisfies the local bisection hypothesis. Westart with some technical results. Throughout, we will identify Σ ′ (0) with ultrafilters of I ( B ), where B := A R ( G (0) ; q − ( G (0) )), via the homeomorphism described in Remark 5.1. Lemma 8.1.
Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ) , and let B := A R ( G (0) ; q − ( G (0) )) . If U is an ultrafilter of I ( B ) and n ∈ N ( B ) satisfies n † n ∈ U , thenthe upclosure V := ( nU ) ↑ of nU ⊆ N ( B ) \{ } is an ultrafilter of N ( B ) containing n .Proof. Regarding U as an ultrafilter in Σ ′ (0) , we have U ∈ V n † n = s ( V n ), and so there existsan ultrafilter V ∈ V n with s ( V ) = U . Then [5, Lemma 3.1(a)] implies that V = ( nU ) ↑ .Since n † n ∈ U , we have n = nn † n ∈ nU ⊆ V ⊆ N ( B ) \{ } . (cid:3) Lemma 8.2.
Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ) , and let B := A R ( G (0) ; q − ( G (0) )) . Let X ⊆ Σ be a compact open bisection, and let n ∈ N ( B ) . Then (a) ˜1 X ≤ n if and only if n | X ≡ and q ( X ) = s − ( s ( q ( X ))) ∩ supp G ( n ) ; (b) if n ≤ ˜1 X , then n = ˜1 Y for some compact open bisection Y ⊆ X ; and (c) given a compact open bisection Y ⊆ Σ , we have ˜1 Y ≤ ˜1 X if and only if Y ⊆ X .Proof. For part (a), suppose first that ˜1 X ≤ n ; that is, that ˜1 X = n ˜1 † X ˜1 X = n ˜1 s ( X ) .Proposition 2.11 implies that for all σ ∈ X ,1 = ˜1 X ( σ ) = ( n ˜1 s ( X ) )( σ ) = n ( σ ) 1 s ( X ) ( s ( σ )) = n ( σ ) . The containment q ( X ) ⊆ s − ( s ( q ( X ))) ∩ supp G ( n ) follows from the above equality. For thereverse containment, suppose that σ ∈ Σ satisfies q ( σ ) ∈ s − ( s ( q ( X ))) ∩ supp G ( n ). Then s ( q ( σ )) ∈ s ( q ( X )), and so there exists γ ∈ X such that q ( s ( γ )) = s ( q ( γ )) = s ( q ( σ )) = q ( s ( σ )). This implies that there exists a unique t ∈ R × such that s ( σ ) = t · s ( γ ). By theequality ˜1 X = n ˜1 s ( X ) , we obtain˜1 X ( σ ) = ( n ˜1 s ( X ) )( σ ) = n ( σ ) ˜1 s ( X ) ( s ( σ )) = t − n ( σ ) = 0 . This implies that σ = u · σ ′ for some σ ′ ∈ X and u ∈ R × . Hence q ( σ ) = q ( σ ′ ) ∈ q ( X ),and the containment s − ( s ( q ( X ))) ∩ supp G ( n ) ⊆ q ( X ) follows.For the converse, notice that for γ ∈ Σ, applying Proposition 2.11 at the first step gives( n ˜1 s ( X ) )( γ ) = 0 ⇐⇒ n ( γ ) ˜1 s ( X ) ( s ( γ )) = 0 ⇐⇒ γ ∈ supp( n ) and s ( q ( γ )) = q ( s ( γ )) ∈ q ( s ( X )) = s ( q ( X )) ⇐⇒ q ( γ ) ∈ s − ( s ( q ( X ))) ∩ supp G ( n ) = q ( X ) ⇐⇒ γ ∈ R × · X. Since ( n ˜1 s ( X ) )( σ ) = 1 for all σ ∈ X , we deduce that n ˜1 s ( X ) = ˜1 X by Lemma 2.7, andhence ˜1 X ≤ n . For part (b), let Y := X ∩ s − ( s (supp( n ))). Note that s (supp( n )) = s (supp G ( n ))(under the usual identification of G (0) and Σ (0) ) is compact and open, and hence is clopensince G (0) is Hausdorff. Thus Y is a clopen subset of the compact open bisection X , andso Y is a compact open bisection. Since n ≤ ˜1 X by hypothesis, Remark 4.4 implies that n = ˜1 X n † n = ˜1 X ˜1 s (supp( n )) = ˜1 Y . For part (c), let Y be a compact open bisection of Σ. If ˜1 Y ≤ ˜1 X , then since ˜1 Y ∈ N ( B ), part (b) implies that ˜1 Y = ˜1 Z for some compact open bisection Z ⊆ X . Thisforces Y = Z , so that Y ⊆ X . On the other hand, if Y ⊆ X , then X s ( Y ) = Y , andso ˜1 Y = ˜1 X ˜1 s ( Y ) ≤ ˜1 X by Lemma 2.17(d) and Corollary 2.10 (since ˜1 s ( Y ) ∈ I ( B ) and˜1 X ∈ N ( B )). (cid:3) Lemma 8.3.
Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ) , and let B := A R ( G (0) ; q − ( G (0) )) . Fix σ ∈ Σ . The set U s ( σ ) := { e ∈ I ( B ) : e ( s ( σ )) = 1 } is an ultrafilterof I ( B ) . (a) If X and Y are compact open bisections such that σ ∈ X ∩ Y , then (˜1 X U s ( σ ) ) ↑ =(˜1 Y U s ( σ ) ) ↑ . (b) If X and Y are compact open bisections such that σ ∈ X and ˜1 Y ∈ (˜1 X U s ( σ ) ) ↑ ,then σ ∈ Y . (c) If X is a compact open bisection such that σ ∈ X and n ∈ (˜1 X U s ( σ ) ) ↑ , then thereexists a compact open set W ⊆ s ( X ) such that σ ∈ XW and ˜1 XW ≤ n .Proof. A routine argument shows that U s ( σ ) is an ultrafilter of I ( B ).For part (a), it suffices by symmetry to show that (˜1 X U s ( σ ) ) ↑ ⊆ (˜1 Y U s ( σ ) ) ↑ . Since theright-hand side is closed upwards, it suffices to show that it contains ˜1 X U s ( σ ) . But theright-hand side is also closed under right multiplication by U s ( σ ) . Thus it suffices to showthat ˜1 X ∈ (˜1 Y U s ( σ ) ) ↑ . Since X ∩ Y is open, there is a compact open set W ⊆ X ∩ Y with σ ∈ W , and clearly W is a compact open bisection. So ˜1 W ≤ ˜1 X , ˜1 Y by Lemma 8.2(c),and ˜1 s ( W ) ∈ U s ( σ ) since σ ∈ W . Note that ˜1 W = ˜1 Y ˜1 s ( W ) by Corollary 2.10. Therefore,˜1 Y ˜1 s ( W ) = ˜1 W ≤ ˜1 X , and so ˜1 X ∈ (˜1 Y U s ( σ ) ) ↑ , as required.For part (b), note that since ˜1 Y ∈ (˜1 X U s ( σ ) ) ↑ , there exists a compact open set W ⊆ Σ (0) such that ˜1 W ∈ U s ( σ ) and ˜1 XW = ˜1 X ˜1 W ≤ ˜1 Y by Corollary 2.10. Thus σ = σ s ( σ ) ∈ XW ,and so ˜1 Y ( σ ) = 1 by Lemma 8.2(a). Hence σ ∈ Y .For part (c), note that since n ∈ (˜1 X U s ( σ ) ) ↑ , there exists a compact open set W ⊆ Σ (0) such that ˜1 W ∈ U s ( σ ) and ˜1 X ˜1 W ≤ n . By replacing W with W ∩ s ( X ), we may assumethat W ⊆ s ( X ). Hence ˜1 XW = ˜1 X ˜1 W ≤ n . Since W ∈ U s ( σ ) , we have s ( σ ) ∈ W , and so σ = σ s ( σ ) ∈ XW . (cid:3) We are now in a position to show that for any discrete R -twist (Σ , i, q ) over G , thereis a continuous open groupoid homomorphism Φ of Σ into the twist obtained from thepair ( A R ( G ; Σ) , A R ( G (0) ; q − ( G (0) ))), and that this induces a morphism of twists. We firstestablish the existence of Φ. Proposition 8.4.
Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ) , andlet B := A R ( G (0) ; q − ( G (0) )) . Let Σ ′ be the groupoid of ultrafilters of N ( B ) defined inSection 5.1, and let G ′ be the corresponding quotient of Σ ′ by the action of R × . For each x ∈ Σ (0) , let U x be the ultrafilter { e ∈ I ( B ) : e ( x ) = 1 } of I ( B ) corresponding to x . For σ ∈ Σ , let X be a compact open bisection containing σ , and let Φ( σ ) := (˜1 X U s ( σ ) ) ↑ ∈ Σ ′ . ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 37
Then Φ( σ ) does not depend on the choice of X . The map Φ : Σ → Σ ′ is an R × -equivariantcontinuous open embedding of topological groupoids such that Φ(Σ (0) ) = Σ ′ (0) .Proof. Lemma 8.3(a) shows that Φ( σ ) does not depend on the choice of X . To see thatΦ is injective, suppose that σ and τ are distinct elements of Σ. Since Σ is Hausdorff,there is a compact open bisection X ⊆ Σ such that σ ∈ X and τ / ∈ X . In particular,˜1 X ˜1 s ( X ) = ˜1 X ∈ Φ( σ ) \ Φ( τ ) by Corollary 2.10 and Lemma 8.3(b), and so Φ( σ ) = Φ( τ ).Fix x ∈ Σ (0) . By taking any compact open neighbourhood X ⊆ Σ (0) of x , we seethat Φ( x ) = (˜1 X U x ) ↑ = ( U x ) ↑ is the unique ultrafilter of N ( B ) containing U x . By Stoneduality, it follows that Φ restricts to a homeomorphism from Σ (0) to Σ ′ (0) .Fix σ ∈ Σ and let X ⊆ Σ be a compact open bisection containing σ . Put V :=(˜1 X U s ( σ ) ) ↑ = Φ( σ ). Then s (Φ( σ )) = V − V . If e ∈ U s ( σ ) , then e ≥ (˜1 X e ) † (˜1 X e ) ∈ V − V ,and so e ∈ V − V . Therefore, Φ( s ( σ )) = ( U s ( σ ) ) ↑ ⊆ V − V , and so by the maximalityproperty of ultrafilters, we have Φ( s ( σ )) = V − V = s (Φ( σ )). Suppose that e ∈ U r ( σ ) ⊆ Φ( r ( σ )). Then (˜1 † X e ˜1 X )( s ( σ )) = 1 by Proposition 2.11, and so ˜1 † X e ˜1 X ∈ U s ( σ ) = V − V .Therefore, e ≥ ˜1 X (˜1 † X e ˜1 X )˜1 † X ∈ V ( V − V ) V − = V V − , and so e ∈ V V − . ThusΦ( r ( σ )) = ( U r ( σ ) ) ↑ ⊆ V V − = r (Φ( σ )), and so Φ( r ( σ )) = r (Φ( σ )) since these areultrafilters. It follows that Φ carries composable pairs to composable pairs.To see that Φ is multiplicative, fix ( σ, τ ) ∈ Σ (2) , and choose a compact open bisection W ⊆ Σ containing στ . By the continuity of multiplication, there exist compact openbisections X and Y of Σ such that σ ∈ X , τ ∈ Y , and XY ⊆ W . Then ˜1 X ˜1 Y = ˜1 XY ≤ ˜1 W by Corollary 2.10 and Lemma 8.2(c). It follows that ˜1 W ∈ Φ( σ )Φ( τ ). Since s ( τ ) = s ( στ ),it follows that ˜1 W U s ( στ ) ⊆ Φ( σ )Φ( τ ) U s ( τ ) ⊆ Φ( σ )Φ( τ ), and so Φ( στ ) ⊆ Φ( σ )Φ( τ ). HenceΦ( στ ) = Φ( σ )Φ( τ ) by the maximality property of ultrafilters.To see that Φ is continuous, fix n ∈ N ( B ), and let σ ∈ Φ − ( V n ). Choose a compact openbisection X ⊆ Σ containing σ . Then (˜1 X U s ( σ ) ) ↑ = Φ( σ ) ∈ V n , and so by the definition of V n , we have n ∈ (˜1 X U s ( σ ) ) ↑ . It follows from Lemma 8.3(c) that there is a compact openset W ⊆ s ( X ) such that σ ∈ XW and ˜1 XW ˜1 s ( X ) = ˜1 XW ≤ n . Hence Φ( τ ) ∈ V n for each τ ∈ XW , and so XW is an open neighbourhood of σ contained in Φ − ( V n ).To see that Φ is open, it suffices to fix a compact open bisection X of Σ, and showthat Φ( X ) = V ˜1 X . If σ ∈ X , then certainly ˜1 X ∈ Φ( σ ), and so Φ( σ ) ∈ V ˜1 X . This givesΦ( X ) ⊆ V ˜1 X . For the reverse containment, suppose that U ∈ V ˜1 X . Then ˜1 X ∈ U , and so˜1 s ( X ) = ˜1 † X ˜1 X ∈ s ( U ). We already know that s ( U ) = ( U x ) ↑ for some x ∈ Σ (0) , and hence(˜1 s ( X ) )( x ) = 1, which implies that x ∈ s ( X ). Let σ be the unique element of X such that s ( σ ) = x . We claim that U = Φ( σ ). Indeed, since s ( U ) = ( U x ) ↑ = ( U s ( σ ) ) ↑ and ˜1 X ∈ U ,we have ˜1 X U s ( σ ) ⊆ U , and hence Φ( σ ) ⊆ U . Thus U = Φ( σ ) ⊆ Φ( X ) by the maximalityproperty of ultrafilters, and so Φ is an open map.It remains to prove R × -equivariance. Fix t ∈ R × and σ ∈ Σ. Fix a compact openbisection X ⊆ Σ containing σ . Then t · X is a compact open bisection containing t · σ ,and ˜1 t · X = t ˜1 X . Since s ( t · σ ) = s ( σ ), we have Φ( t · σ ) = (˜1 t · X U s ( σ ) ) ↑ . Now˜1 t · X U s ( σ ) = ( t ˜1 X ) U s ( σ ) ⊆ t · Φ( σ ) , and thus Φ( t · σ ) ⊆ t · Φ( σ ). Hence Φ( t · σ ) = t · Φ( σ ) since they are ultrafilters. (cid:3) It is now relatively straightforward to show that Φ induces a morphism of discrete R -twists. Corollary 8.5.
Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ) , and let B := A R ( G (0) ; q − ( G (0) )) . Let G ′ (0) × R × i ′ ֒ → Σ ′ q ′ ։ G ′ be the sequence obtained from ( A, B ) via Theorem 5.6. Let Φ : Σ → Σ ′ be the map ofProposition 8.4. Then there is a (well-defined) map Φ G : G → G ′ given by Φ G ( q ( σ )) = q ′ (Φ( σ )) , and the following diagram commutes. G (0) × R × Σ GG ′ (0) × R × Σ ′ G ′ i Φ G × id q Φ Φ G i ′ q ′ Proof.
Proposition 8.4 shows that Φ is R × -equivariant. Thus Φ G is well-defined, and theright-hand square of the diagram commutes. To see that the left-hand square commutes,observe that for all x ∈ Σ (0) , we haveΦ( i ( q ( x ) , t )) = Φ( t · x ) = t · Φ( x ) = i ′ ( q ′ (Φ( x )) , t ) = i ′ (Φ G ( q ( x )) , t ) . (cid:3) To proceed, we need the following observation about normalisers n ∈ N ( B ) withsupp G ( n ) a bisection. Proposition 8.6.
Fix n ∈ N ( B ) . Then supp G ( n ) is a bisection if and only if n = ˜1 X fora (necessarily unique) compact open bisection X of Σ .Proof. Suppose that X ⊆ Σ is a compact open bisection of Σ. Then supp G (˜1 X ) = q ( X )is a compact open bisection of G , since q is a continuous open map that restricts to ahomeomorphism of unit spaces. Moreover, ˜1 − X (1) = X , and so uniqueness is clear.For the converse, suppose that supp G ( n ) is a compact open bisection of G . Note that n † n = ˜1 s (supp( n )) by Remark 4.4. Fix σ ∈ supp( n ). Then 1 = ( n † n )( s ( σ )) = n † ( σ − ) n ( σ )since supp G ( n ) is a bisection, and so n ( σ ) ∈ R × . It follows that n takes values in R × ,and so by the R × -contravariance of n , we have supp( n ) = R × · X , where X := n − (1).Since n is continuous, X is open. Moreover, q ( X ) = supp G ( n ) by the above observation.We claim that q | X is injective. To see this, suppose that q ( σ ) = q ( τ ) for σ, τ ∈ X . Then τ = t · σ for some t ∈ R × , and so n ( σ ) = 1 = n ( τ ) = t − . Hence t = 1, and so σ = τ .Therefore, X is homeomorphic to q ( X ) = supp G ( n ), and hence X is compact. To see that X is a bisection, note that if s ( σ ) = s ( τ ) for σ, τ ∈ X , then q ( σ ) = q ( τ ) since supp G ( n )is a bisection, and so σ = τ by the injectivity of q | X . Similarly, r | X is injective. Since ˜1 X and n are supported on R × · X and agree on X , we deduce that n = ˜1 X by Lemma 2.7. (cid:3) It follows from Proposition 8.6 that X ˜1 X is an isomorphism of the inverse semi-group of compact open bisections of Σ onto N ( B ) if the twist Σ → G satisfies the localbisection hypothesis (using Corollary 2.10). Therefore, we deduce by so-called noncom-mutative Stone duality [31] that Σ ∼ = Σ ′ . Moreover, the map Φ is the isomorphism thatnoncommutative Stone duality provides. The next result gives a more precise statement;namely, that the map Φ of Proposition 8.4 is surjective precisely when the pair of algebrasassociated to G is an algebraic quasi-Cartan pair. Theorem 8.7.
Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ) , and let B := A R ( G (0) ; q − ( G (0) )) . Let Σ ′ be the groupoid of ultrafilters of N ( B ) defined in Section 5.1.Let Φ : Σ → Σ ′ be the map from Proposition 8.4. The following are equivalent. (1) The pair ( A, B ) is an algebraic quasi-Cartan pair. (2) The twist (Σ , i, q ) satisfies the local bisection hypothesis. (3) The map Φ is surjective and, in particular, it is an isomorphism of topologicalgroupoids.Proof. Lemma 4.2 gives (1) ⇐⇒ (2).To see that (2) = ⇒ (3), let U ∈ Σ ′ and n ∈ U . Then supp G ( n ) is a bisection of G , andso by Proposition 8.6, there is a compact open bisection X of Σ such that n = ˜1 X . For ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 39 each x ∈ Σ (0) , let U x be the ultrafilter { e ∈ I ( B ) : e ( x ) = 1 } of I ( B ). By Proposition 8.4,we have s ( U ) = ( U x ) ↑ for some x ∈ Σ (0) , and since ˜1 s ( X ) = n † n ∈ s ( U ), there exists e ∈ U x such that e ≤ ˜1 s ( X ) . Since e ( x ) = 1, Lemma 8.2(c) implies that x ∈ s ( X ). Since X is abisection, there is a unique σ ∈ X with s ( σ ) = x . Note that ˜1 X U x = nU x ⊆ U s ( U ) = U ,and hence Φ( σ ) = (˜1 X U s ( σ ) ) ↑ ⊆ U . Thus U = Φ( σ ) since these are ultrafilters. Therefore,Φ is surjective and hence is an isomorphism by Proposition 8.4.To see that (3) = ⇒ (2), fix n ∈ N ( B ) \{ } and x ∈ s (supp G ( n )). Note that n † n =˜1 s (supp( n )) by Remark 4.4, and hence s (supp G ( n )) = supp G ( n † n ). Therefore, n † n ∈ U x , andso Lemma 8.1 implies that ( nU x ) ↑ is an ultrafilter of N ( B ) containing n , with source U x .Since Φ is surjective and respects the source map, there exists τ ∈ Σ with s ( τ ) = x suchthat Φ( τ ) = ( nU x ) ↑ . So n ∈ Φ( τ ). Let X be a compact open bisection of Σ containing τ ,so that Φ( τ ) = (˜1 X U s ( τ ) ) ↑ . By Lemma 8.3(c), there exists a compact open set W ⊆ s ( X )such that τ ∈ XW and ˜1 XW ≤ n . Since XW ⊆ X is a bisection, Lemma 8.2(a) impliesthat q ( τ ) is the unique element of supp G ( n ) with s ( q ( τ )) = x . Since x ∈ s (supp G ( n )) wasarbitrary, it follows that s is injective on supp G ( n ). A symmetric argument shows that r is injective on supp G ( n ), and so supp G ( n ) is a bisection. (cid:3) To finish off, we deduce that the local bisection hypothesis is encoded algebraicallyin the sense that if the algebraic pairs corresponding to two twists are isomorphic, theneither neither of the two twists satisfies the local bisection hypothesis, or they both do.
Definition 8.8.
Let (Σ , i , q ) and (Σ , i , q ) be discrete R -twists over ample Haus-dorff groupoids G and G , respectively. We say that an isomorphism Ψ : A R ( G ; Σ ) → A R ( G ; Σ ) is diagonal-preserving if Ψ (cid:0) A R ( G (0)1 ; q − ( G (0)1 )) (cid:1) = A R ( G (0)2 ; q − ( G (0)2 )). Remark . Let (Σ , i, q ) be a discrete R -twist over G , let A := A R ( G ; Σ), and let B := A R ( G (0) ; q − ( G (0) )). As in the untwisted case, we call B the diagonal subalgebra of A ,even though Proposition 7.1 says that ( A, B ) is an algebraic diagonal pair if and only if G is principal. Corollary 8.10.
Let (Σ , i , q ) and (Σ , i , q ) be discrete R -twists over ample Hausdorffgroupoids G and G , respectively. Suppose that Σ → G satisfies the local bisectionhypothesis. The following are equivalent. (1) The twists (Σ , i , q ) and (Σ , i , q ) are isomorphic. (2) There exists a diagonal-preserving isomorphism of R -algebras Ψ : A R ( G ; Σ ) → A R ( G ; Σ ) .Proof. The implication (1) = ⇒ (2) is immediate. We prove the implication (2) = ⇒ (1).Write ( A , B ) and ( A , B ) for the algebraic pairs associated with Σ → G and Σ → G , respectively. We first show that ( A , B ) is an algebraic quasi-Cartan pair. Define˜ P : A → B by ˜ P ( a ) = Ψ( P (Ψ − ( a ))), where P is the unique faithful conditionalexpectation satisfying (AQP) for the pair ( A , B ). Since Ψ is diagonal-preserving, it isstraightforward to prove that ˜ P | B = id B , that ˜ P ( b a b ′ ) = b ˜ P ( a ) b ′ for all a ∈ A and b , b ′ ∈ B , and that Ψ( N ( B )) = N ( B ). The latter implies that ˜ P is faithful and thatit satisfies condition (AQP). Since Ψ and P are R -linear, ˜ P is R -linear. Hence the pair( A , B ) with ˜ P is an algebraic quasi-Cartan pair. By Lemma 4.2, (Σ , i , q ) satisfies thelocal bisection hypothesis, and by Proposition 3.7 and Lemma 4.2, ˜ P coincides with theconditional expectation of ( A , B ) given by restriction of functions.We now build an isomorphism between the twists. By Theorem 8.7(3) andCorollary 8.5, we can identify each Σ i with the groupoid of ultrafilters in N ( B i ), andwe can identify each G i with the corresponding quotient. Since the partial order oneach N ( B i ) is defined in terms of the multiplication, it follows that if U is a filter of N ( B ), then Ψ( U ) is a filter of N ( B ), and if V is a filter of N ( B ), then Ψ − ( V ) is afilter of N ( B ). Hence the map ψ : Σ → Σ given by ψ ( U ) := Ψ( U ) is a well-definedbijection. Moreover, since the multiplication and topology on each Σ i depend only on themultiplicative structure of N ( B i ), it is straightforward to show that ψ is a topologicalgroupoid isomorphism. Since Ψ is R -linear, we have ψ ( tU ) = tψ ( U ) for all t ∈ R × and U ∈ Σ , and so ψ is R × -equivariant. Hence there is a (well-defined) topological groupoidisomorphism ψ G : G → G given by ψ G ( q ( U )) := q ( ψ ( U )). It follows that the diagram G (0)1 × R × Σ G G (0)2 × R × Σ G i ψ G × id q ψ ψ G i q gives an isomorphism between of the twists (Σ , i , q ) and (Σ , i , q ). (cid:3) If at least one of the two twists is effective, then we can strengthen the previous result byasking only for an isomorphism of twisted Steinberg algebras that restricts to an inclusionof diagonal subalgebras; this implies that in fact it restricts to an isomorphism of diagonalsubalgebras.
Corollary 8.11.
Let (Σ , i , q ) and (Σ , i , q ) be discrete R -twists over ample Haus-dorff groupoids G and G , respectively. Suppose that G is effective. The following areequivalent. (1) The twists (Σ , i , q ) and (Σ , i , q ) are isomorphic. (2) There exists a diagonal-preserving isomorphism of R -algebras Ψ : A R ( G ; Σ ) → A R ( G ; Σ ) . (3) There exists an isomorphism of R -algebras Ψ : A R ( G ; Σ ) → A R ( G ; Σ ) suchthat Ψ (cid:0) A R ( G (0)1 ; q − ( G (0)1 )) (cid:1) ⊆ A R ( G (0)2 ; q − ( G (0)2 )) .Proof. Since G is effective, Lemmas 4.2 and 3.6 and Proposition 4.8 together imply that(Σ , i , q ) satisfies the local bisection hypothesis. The equivalence (1) ⇐⇒ (2) followsfrom Corollary 8.10. The implication (2) = ⇒ (3) is clear. We prove that (3) = ⇒ (2).Let Ψ be an isomorphism as in (3). Since G is effective, Proposition 4.8 implies that B := A R ( G (0)1 ; q − ( G (0)1 )) is a maximal abelian subalgebra of A R ( G ; Σ ), which impliesthat Ψ( B ) is a maximal abelian subalgebra of A R ( G ; Σ ). Since A R ( G (0)2 ; q − ( G (0)2 )) isan abelian subalgebra containing Ψ( B ), they are equal. (cid:3) Examples satisfying the local bisection hypothesis
In this short section we show that our theorems apply to a relatively large class ofgroupoids and twists. In particular, all twists over Deaconu–Renault groupoids withtotally disconnected unit spaces satisfy the local bisection hypothesis.The starting point is to show that there are plenty of groups whose twisted groupalgebras over large classes of indecomposable rings have no nontrivial units. Most ofwhat we write here is well known.Recall that a group G is right orderable if there is a total order < on G such that x < y implies that xz < yz for all x, y, z ∈ G . If, in addition, the order satisfies zx < zy for all x, y, z ∈ G , then G is said to be orderable . Note that a group G is (right) orderable if andonly if all of its finitely generated subgroups are too [41, Lemma 13.2.2].A group G is said to have the unique product property if, given two finite nonemptysubsets A and B of G , there is an element of AB that can be uniquely written as a ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 41 product ab with a ∈ A and b ∈ B . This property is satisfied by any right-orderable group[41, Theorem 13.1.7] and so, in particular, by any torsion-free abelian group, torsion-freenilpotent group [41, Lemma 13.1.6], and free group. Note that locally indicable groups areright orderable. Here, a group G is locally indicable if every finitely generated subgroupadmits a nontrivial homomorphism to an infinite cyclic group.It was shown by Strojnowski [48] that if G has the unique product property and A and B are finite subsets of G with | A | + | B | >
2, then there are two distinct elements g, h ∈ AB which have unique representations g = ab and h = a ′ b ′ with a, a ′ ∈ A and b, b ′ ∈ B . Notethat every group satisfying the unique product property is torsion-free, and that there aregroups satisfying the unique product property that are not right orderable.If R is an integral domain and G has the unique product property, then it is wellknown that the group ring RG has no nontrivial units. The same argument applies mutatis mutandis to twisted group rings. A more general result for group graded ringscan be found in [40, Theorem 3.4]. We include the elementary proof for completeness. Lemma 9.1.
Let G be a group satisfying the unique product property, let R be a unitalintegral domain, and let c be a normalised R × -valued -cocycle on G . Then the only unitsof the twisted group ring R ( G, c ) are the elements of the set { tδ g : t ∈ R × , g ∈ G } .Proof. Let e denote the identity of G . Suppose that a, b ∈ R ( G, c ) satisfy ab = ba =1 R ( G,c ) = δ e . Write a = P g ∈ G a g δ g and b = P h ∈ G b h δ h . Let A := { g : a g = 0 } and B := { h : b h = 0 } . These are finite sets. If a and b are not both trivial units, then | A | + | B | >
2. So there exist distinct g, h ∈ AB which have unique factorisations g = xy and h = x ′ y ′ with x, x ′ ∈ A and y, y ′ ∈ B . It follows that the coefficients of δ g and δ h in ab are a x b y c ( x, y ) and a x ′ b y ′ c ( x ′ , y ′ ), respectively, and these are both nonzero since R is anintegral domain. This contradicts ab = δ e . Therefore, | A | + | B | ≤
2, and so a = a g δ g and b = b h δ h with g, h ∈ G . We then have a g b h c ( g, h ) = 1, and so a g , b h ∈ R × , as required. (cid:3) We call G a trivial-units-only group if the twisted group ring k ( G, c ) has no nontrivialunits for every field k and every normalised k × -valued 2-cocycle c on G . For example, agroup satisfying the unique product property is a trivial-units-only group by Lemma 9.1.Recall that a commutative unital ring R is reduced if it has no nonzero nilpotentelements. This is equivalent to the intersection of all prime ideals of R being trivial.Every integral domain is reduced and indecomposable. However, the class of reduced andindecomposable commutative rings is much larger and includes the coordinate rings ofconnected (but not necessarily irreducible) affine varieties over algebraically closed fields.For instance, C [ x, y ] / ( xy ), which is the coordinate ring of the union of the coordinate axesin C , is reduced and indecomposable but not an integral domain.The following proposition is a variation of [47, Proposition 2.1], which is based on [38,Theorem 3]. It generalises Lemma 9.1. Proposition 9.2.
Let G be a nontrivial trivial-units-only group as defined above (forexample, a group satisfying the unique product property). Let R be a commutative unitalring and let c be an R × -valued normalised -cocycle on G . Then R ( G, c ) has no nontrivialunits if and only if R is reduced and indecomposable.Proof. Let e denote the identity of G . We begin with necessity; this argument appliesto all nontrivial groups G . Fix g ∈ G \{ e } . Suppose first that R is not indecomposable,and let f ∈ R \{ , } be an idempotent. Then a = f δ e + (1 − f ) δ g is a nontrivial unitwith inverse b = f δ e + (1 − f ) c ( g, g − ) − δ g − . Indeed, using that c is normalised, we havethat ab = (cid:0) f + (1 − f ) c ( g, g − ) c ( g, g − ) − (cid:1) δ e = δ e , and similarly, ba = δ e , using that c ( g, g − ) = c ( g − , g ), since c is a normalised 2-cocycle. Next assume that R is not reduced. If n ∈ R \{ } is nilpotent, then nδ g is nilpotent,and so δ e − nδ g is a nontrivial unit with inverse δ e + nδ g .For sufficiency, we use some algebraic geometry, following [38]. Since R is indecompos-able, its prime (Zariski) spectrum Spec( R ) is connected. If p ∈ Spec( R ), then the residuefield at p is the field of fractions κ ( p ) of R/ p . For each r ∈ R , we write r ( p ) for the imageof r in κ ( p ) under the composition R → R/ p → κ ( p ). We have an induced κ ( p ) × -valuednormalised 2-cocycle c p on G obtained by putting c p ( g, h ) = c ( g, h )( p ). Moreover, thereis a canonical surjective homomorphism π p : R ( G, c ) → κ ( p )( G, c p ) sending P g ∈ G a g δ g to a ( p ) := P g ∈ G a g ( p ) δ g .Let a ∈ R ( G, c ) be a unit, say a = P g ∈ G a g δ g . Then a ( p ) is a unit of κ ( p )( G, c p ) andhence is a trivial unit by the hypothesis that G is a trivial-units-only group. It followsthat there is a unique element h ∈ G with a h ( p ) = 0. Denote this element by g ( p ). Then g : p g ( p ) is a map from Spec( R ) to G . We claim that g is locally constant. Indeed,if g ( p ) = h , then a h / ∈ p . Let D ( a h ) denote the basic open set of all prime ideals notcontaining a h . For q ∈ D ( a h ), we have a h ( q ) = 0, and so g ( q ) = h . Therefore, g isconstant on D ( a h ), and so g is locally constant, as claimed. Since Spec( R ) is connected,we deduce that g is constant. Hence there exists h ∈ G such that a h ( p ) = 0 for all p ∈ Spec( R ); and for all h ′ = h , we have a h ′ ( p ) = 0 for all p ∈ Spec( R ). Since R isreduced, the intersection of all of its prime ideals is zero, and so a h ′ = 0 for h ′ = h . And,since a h belongs to no prime ideal, and hence to no maximal ideal, it is a unit. Thus a = a h δ h is a trivial unit. (cid:3) Corollary 9.3.
Let R be a reduced and indecomposable commutative ring, and let (Σ , i, q ) be a discrete R -twist over G . Let I denote the interior of the isotropy of G , and supposethat there is a dense set X ⊆ G (0) such that for each x ∈ X , the group I x satisfiesthe unique product property (for example, it might be a right-orderable group). Let A := A R ( G ; Σ) and B := A R ( G (0) ; q − ( G (0) )) ⊆ A . Then ( A, B ) is an algebraic quasi-Cartanpair, and the map Φ of Proposition 8.4 from Σ to the twist Σ ′ is an isomorphism of twists.Proof. Lemma 9.1 and Proposition 9.2 show that twisted group rings of groups satisfyingthe unique product property have no nontrivial units, and so the hypothesis of Lemma 4.6is satisfied. Therefore, Lemma 4.6 and Lemma 4.7(b) together show that (Σ , i, q ) satisfiesthe local bisection hypothesis. The result follows from Corollary 8.5 and Theorem 8.7. (cid:3)
Since free abelian groups are orderable and hence have the unique product property,we have the following corollary of Corollary 9.3.
Corollary 9.4.
Let X be a totally disconnected locally compact Hausdorff space, and let T : n T n be an action of N k on X by local homeomorphisms. Let G be the associatedDeaconu–Renault groupoid, let R be a reduced and indecomposable commutative ring, andlet (Σ , i, q ) be a discrete R -twist over G . Let ( A, B ) := ( A R ( G ; Σ) , A R ( G (0) ; q − ( G (0) ))) be the associated algebraic pair. Then ( A, B ) is an algebraic quasi-Cartan pair, andTheorem 8.7 recovers the groupoid G and the discrete twist Σ from ( A, B ) .Example . Let Λ be a countable row-finite higher-rank graph with no sources as in[26], let R be a reduced and indecomposable commutative ring, and let ω be normalised R × -valued 2-cocycle on Λ as in [27]. The arguments of [27, Section 6] show how toconstruct a normalised locally constant R × -valued 2-cocycle c ω on the k -graph groupoid G Λ . Since this groupoid is a Deaconu–Renault groupoid, our results show that the twist G Λ × c ω R × over G Λ obtained from c ω can be recovered from the twisted Steinberg alge-bra A R ( G Λ ; G Λ × c ω R × ) and its canonical abelian subalgebra (isomorphic to) C c (Λ ∞ , R ). ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 43
Moreover, A R ( G Λ ; G Λ × c ω R × ) ∼ = A R ( G Λ , c ω ) as in [4, Theorem 4.23], and one can re-cover the cohomology class of c ω from the twist G Λ × c ω R × over G Λ as in [4, Corol-lary 4.16]. So the groupoid G Λ and the cohomology class of c ω are invariants of the pair( A R ( G Λ , c ω ) , C c (Λ ∞ , R )). References [1] G. Abrams and G. Aranda Pino,
The Leavitt path algebra of a graph , J. Algebra (2005), 319–334.MR 2172342[2] P. Ara, J. Bosa, R. Hazrat, and A. Sims,
Reconstruction of graded groupoids from graded Steinbergalgebras , Forum Math. (2017), 1023–1037. MR 3692025[3] P. Ara, M.A. Moreno, and E. Pardo, Nonstable K -theory for graph algebras , Algebr. Represent.Theory (2007), 157–178. MR 2310414[4] B. Armstrong, L.O. Clark, K. Courtney, Y.-F. Lin, K. McCormick, and J. Ramagge, Twisted Stein-berg algebras , preprint, 2020, arXiv:1910.13005v2 [math.RA].[5] B. Armstrong, L.O. Clark, A. an Huef, M. Jones, and Y.-F. Lin,
Filtering germs: groupoids associatedto inverse semigroups , preprint, 2020, arXiv:2010.16113v1 [math.RA].[6] S.K. Berberian,
Baer ∗ -rings , Springer-Verlag, New York-Berlin, 1972, Die Grundlehren der mathe-matischen Wissenschaften, Band 195. MR 0429975[7] T. Bice, Representing rings on ringoid bundles , preprint, 2020, arXiv:2012.03006v1 [math.RA].[8] T. Bice and C. Starling,
General non-commutative locally compact locally Hausdorff Stone duality ,Adv. Math. (2019), 40–91. MR 3872844[9] J.H. Brown, L.O. Clark, C. Farthing, and A. Sims,
Simplicity of algebras associated to ´etale groupoids ,Semigroup Forum (2014), 433–452.[10] J.H. Brown, L.O. Clark, and A. an Huef, Diagonal-preserving ring ∗ -isomorphisms of Leavitt pathalgebras , J. Pure Appl. Algebra (2017), 2458–2481. MR 3646311[11] N. Brownlowe, T.M. Carlsen, and M.F. Whittaker, Graph algebras and orbit equivalence , ErgodicTheory Dynam. Systems (2017), 389–417. MR 3614030[12] T.M. Carlsen, ∗ -isomorphism of Leavitt path algebras over Z , Adv. Math. (2018), 326–335.MR 3733888[13] T.M. Carlsen and J. Rout, Diagonal-preserving graded isomorphisms of Steinberg algebras , Commun.Contemp. Math. (2018), 1–25. MR 3848066[14] T.M. Carlsen, E. Ruiz, A. Sims, and M. Tomforde, Reconstruction of groupoids and C*-rigidity ofdynamical systems , Adv. Math., to appear, arXiv:1711.01052v1 [math.OA].[15] L.O. Clark, C. Farthing, A. Sims, and M. Tomforde,
A groupoid generalization of Leavitt pathalgebras , Semigroup Forum (2014), 501–517.[16] L.O. Clark and R. Hazrat, ´etale groupoids and steinberg algebras, a concise introduction , preprint,2019, arXiv:1901.01612v1 [math.RA].[17] L.O. Clark and A. Sims, Equivalent groupoids have Morita equivalent Steinberg algebras , J. PureAppl. Algebra (2015), no. 6, 2062–2075. MR 3299719[18] R. Exel,
Reconstructing a totally disconnected groupoid from its ample semigroup , Proc. Amer. Math.Soc. (2010), 2991–3001. MR 2644910[19] J. Feldman and C.C. Moore,
Ergodic equivalence relations, cohomology, and von Neumann algebras ,Bull. Amer. Math. Soc. (1975), 921–924. MR 425075[20] , Ergodic equivalence relations, cohomology, and von Neumann algebras. I , Trans. Amer.Math. Soc. (1977), 289–324. MR 578656[21] ,
Ergodic equivalence relations, cohomology, and von Neumann algebras. II , Trans. Amer.Math. Soc. (1977), 325–359. MR 578730[22] A. an Huef, A. Kumjian, and A. Sims,
A Dixmier–Douady theorem for Fell algebras , J. Funct. Anal. (2011), 1543–1581.[23] I. Kaplansky,
Rings of operators , W.A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778[24] K. Keimel,
Alg`ebres commutatives engendr´ees par leurs ´el´ements idempotents , Canadian J. Math. (1970), 1071–1078. MR 272765[25] A. Kumjian, On C*-diagonals , Canad. J. Math. (1986), 969–1008. MR 854149[26] A. Kumjian and D. Pask, Higher rank graph C*-algebras , New York J. Math. (2000), 1–20.MR 1745529 [27] A. Kumjian, D. Pask, and A. Sims, On twisted higher-rank graph C*-algebras , Trans. Amer. Math.Soc. (2015), no. 7, 5177–5216. MR 3335414[28] M.V. Lawson,
Inverse semigroups , World Scientific Publishing Co., Inc., River Edge, NJ, 1998, Thetheory of partial symmetries. MR 1694900[29] ,
A noncommutative generalization of Stone duality , J. Aust. Math. Soc. (2010), 385–404.MR 2827424[30] , Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras ,Internat. J. Algebra Comput. (2012), 1–47. MR 2974110[31] M.V. Lawson and D.H. Lenz, Pseudogroups and their ´etale groupoids , Adv. Math. (2013), 117–170. MR 3077869[32] K. Matsumoto and H. Matui,
Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras , Kyoto J. Math. (2014), 863–877. MR 3276420[33] , Full groups of Cuntz-Krieger algebras and Higman-Thompson groups , Groups Geom. Dyn. (2017), no. 2, 499–531. MR 3668049[34] H. Matui, Homology and topological full groups of ´etale groupoids on totally disconnected spaces ,Proc. Lond. Math. Soc. (3) (2012), no. 1, 27–56. MR 2876963[35] ,
Topological full groups of one-sided shifts of finite type , J. Reine Angew. Math. (2015),35–84. MR 3377390[36] , ´etale groupoids arising from products of shifts of finite type , Adv. Math. (2016), 502–548.MR 3552533[37] P.S. Muhly and D.P. Williams,
Continuous trace groupoid C*-algebras. II , Math. Scand. (1992),no. 1, 127–145. MR 1174207[38] E. Neher, Invertible and nilpotent elements in the group algebra of a unique product group , ActaAppl. Math. (2009), no. 1, 135–139. MR 2540962[39] V.V. Nekrashevych,
Cuntz–Pimsner algebras of group actions , J. Operator Theory (2004), no. 2,223–249. MR 2119267[40] J. ¨Oinert, Units, zero-divisors and idempotents in rings graded by torsion-free groups , preprint, 2019,arXiv:1904.04847v2 [math.RA].[41] D.S. Passman,
The algebraic structure of group rings , Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 470211[42] A.L.T. Paterson,
Groupoids, inverse semigroups, and their operator algebras , Progress in Mathemat-ics, vol. 170, Birkh¨auser Boston, Inc., Boston, MA, 1999. MR 1724106[43] J. Renault,
Cartan subalgebras in C*-algebras , Irish Math. Soc. Bull. (2008), no. 61, 29–63.MR 2460017[44] W. Sawin,
A set theoretic question arising from trying to understand a sheaf cohomology question ,MathOverflow, URL: https://mathoverflow.net/q/381534 (version: 2021-01-18).[45] B. Steinberg,
A groupoid approach to discrete inverse semigroup algebras , Adv. Math. (2010),no. 2, 689–727. MR 2565546[46] ,
Modules over ´etale groupoid algebras as sheaves , J. Aust. Math. Soc. (2014), no. 3,418–429. MR 3270778[47] , Diagonal-preserving isomorphisms of ´etale groupoid algebras , J. Algebra (2019), 412–439. MR 3873946[48] A. Strojnowski,
A note on u.p. groups , Comm. Algebra (1980), no. 3, 231–234. MR 558112[49] Y. Watatani, Index for C*-subalgebras , Mem. Amer. Math. Soc. (1990), no. 424, vi+117.MR 996807[50] R. Wiegand, Sheaf cohomology of locally compact totally disconnected spaces , Proc. Amer. Math.Soc. (1969), 533–538. MR 253324 ECONSTRUCTION OF TWISTED STEINBERG ALGEBRAS 45 (B. Armstrong and L.O. Clark)
School of Mathematics and Statistics, Victoria Universityof Wellington, PO Box 600, Wellington 6140, NEW ZEALAND
Email address : [email protected], [email protected] (G.G. de Castro) Departamento de Matem´atica, Universidade Federal de Santa Catarina,Florian´opolis, 88040-900, Brazil
Email address : [email protected] (K. Courtney) Mathematical Institute, WWU M¨unster, Einsteinstr. 62, 48149 M¨unster,GERMANY
Email address : [email protected] (Y.-F. Lin) Mathematical Sciences Research Centre, Queen’s University Belfast,Belfast, BT7 1NN, UNITED KINGDOM
Email address : [email protected] (K. McCormick) Department of Mathematics and Statistics, California State University,Long Beach, CA, UNITED STATES
Email address : [email protected] (J. Ramagge) Faculty of Science, Durham University, Durham, DH1 3LE, UNITED KING-DOM
Email address : [email protected] (A. Sims) School of Mathematics and Applied Statistics, University of Wollongong,Northfields Avenue, Wollongong 2522, AUSTRALIA
Email address : [email protected] (B. Steinberg) Department of Mathematics, City College of New York, Convent Avenueat 138th Street, New York, New York 10031, USA.
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