Twisted Steinberg algebras

https://doi.org/10.1016/j.jpaa.2021.106853Get rights and content

Abstract

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units R×, we study the algebra AR(G,σ) consisting of locally constant compactly supported R-valued functions on G, with convolution and involution “twisted” by σ. We also introduce a “discretised” analogue of a twist Σ over a Hausdorff étale groupoid G, and we show that there is a one-to-one correspondence between locally constant 2-cocycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist Σ arising from a locally constant 2-cocycle σ on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra AR(G;Σ), and we show that it coincides with AR(G,σ1). Given any discrete field Fd, we prove a graded uniqueness theorem for AFd(G,σ), and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of AFd(G,σ) is equivalent to minimality of G.

Introduction

Steinberg algebras have become a topic of great interest for algebraists and analysts alike since their independent introduction in [34] and [10]. Before Steinberg algebras were specified by name, they appeared in the details of many constructions of groupoid C*-algebras, such as those in [14], [19], [20], [28]. Not only have these algebras provided useful insight into the analytic theory of groupoid C*-algebras, they give rise to interesting examples of ⁎-algebras; for example, all Leavitt path algebras and Kumjian–Pask algebras can be realised as Steinberg algebras. Moreover, Steinberg algebras have served as a bridge to facilitate the transfer of concepts and techniques between the algebraic and analytic settings; see [5] for one such case.

Thirty years prior to the introduction of Steinberg algebras, Renault [30] initiated the study of twisted groupoid C*-algebras. These are a generalisation of groupoid C*-algebras in which multiplication and involution are twisted by a T-valued 2-cocycle on the groupoid. Twisted groupoid C*-algebras have since proved extremely valuable in the study of structural properties for large classes of C*-algebras. In particular, work of Renault [31], Tu [35], and Barlak and Li [3] has revealed deep connections between twisted groupoid C*-algebras and the UCT problem from the classification program for C*-algebras. For more work on twisted C*-algebras associated to graphs and groupoids, see [2], [4], [11], [17], [18], [21], [22], [23], [24], [25], [33].

Given the success of non-twisted Steinberg algebras and the far-reaching significance of C*-algebraic results relating to twisted groupoid C*-algebras, we expect that a purely algebraic analogue of twisted groupoid C*-algebras will supply several versatile classes of ⁎-algebras to the literature, as well as a new avenue to approach important problems in C*-algebras.

Throughout, let R be a discrete commutative unital ring with units R×. Let Cd denote the ring of complex numbers endowed with the discrete topology, and let Td denote the complex unit circle endowed with the discrete topology. In this article, we introduce the notion of a twisted Steinberg algebra AR(G,σ) constructed from an ample Hausdorff groupoid G and a locally constant R×-valued 2-cocycle σ on G. Our construction generalises the Steinberg algebra AR(G), and provides a purely algebraic analogue of the twisted groupoid C*-algebra C(G,σ) in the case where R=Cd.

In the non-twisted setting, the complex Steinberg algebra and the C*-algebra associated to an ample Hausdorff groupoid G are both built from the convolution algebra Cc(G) of continuous compactly supported complex-valued functions on G. The complex Steinberg algebra A(G) is the ⁎-subalgebra of Cc(G) consisting of locally constant functions, and the full (or reduced) groupoid C*-algebra C(G) (or Cr(G)) is the closure of Cc(G) with respect to the full (or reduced) C*-norm (see [32, Chapter 9]). It turns out (see [10, Proposition 4.2]) that A(G) sits densely inside of both the full and the reduced C*-algebras. Therefore, the definition of a twisted complex Steinberg algebra should result in the same inclusions; that is, the twisted complex involutive Steinberg algebra should sit ⁎-algebraically and densely inside the twisted groupoid C*-algebra. However, to even make sense of that goal, one must first choose between two methods of constructing a twisted groupoid C*-algebra. The first involves twisting the multiplication on C(G) by a continuous T-valued 2-cocycle σ, and was introduced by Renault in [30].

In [30], Renault also observed that the structure of a twisted groupoid C*-algebra with multiplication incorporating a 2-cocycle σ could be realised instead by first twisting the groupoid itself, and then constructing an associated C*-algebra. This is achieved by forming a split groupoid extensionG(0)×TG×σTG, where multiplication and inversion on the groupoid G×σT both incorporate a T-valued 2-cocycle σ on G, and then defining the twisted groupoid C*-algebra to be the completion of the algebra of T-equivariant functions on Cc(G×T) under a C*-norm. A few years later, while developing a C*-analogue of Feldman–Moore theory, Kumjian [18] observed the need for a more general construction arising from a locally split groupoid extensionG(0)×TΣG, where Σ is not necessarily homeomorphic to G×T. It turns out that when G is a second-countable ample Hausdorff groupoid, a folklore result (Theorem 4.10) tells us that every twist over G does arise from a T-valued 2-cocycle on G.

Therefore, our first task is to define twisted Steinberg algebras with respect to both notions of a twist. This is the focus of Sections 3 and 4. In Section 3, we define the twisted Steinberg algebra AR(G,σ) by taking an ample Hausdorff groupoid G and twisting the multiplication of the classical Steinberg algebra AR(G) using a locally constant R×-valued 2-cocycle σ on G. We then show that ACd(G,σ) sits densely inside the twisted groupoid C*-algebra C(G,σ). In Section 4.3, we give an alternative construction of a twisted Steinberg algebra built using a twist Σ over G, and then verify that these two definitions of twisted Steinberg algebras agree when the twist over G arises from a 2-cocycle.

In order to construct a twisted complex Steinberg algebra using a twist over a groupoid, we are forced to first “discretise” our groupoid extension by replacing the standard topology on T with the discrete topology. Though this may seem a little artificial to a C*-algebraist, this change is indeed necessary, as we explain in Remark 4.20. (Nonetheless, this should not come as too much of a surprise, given the purely algebraic nature of Steinberg algebras.) Thus, Section 4.1 is dedicated to introducing these discretised groupoid twists and establishing in this setting the aforementioned folklore result for an arbitrary commutative unital ring R (Theorem 4.10). Then in Section 4.2, we flesh out the relationships between these twists over groupoids and the cohomology theory of groupoids.

Section 5 provides several examples of twisted Steinberg algebras, including a notion of twisted Kumjian–Pask algebras. The final two sections of the paper are devoted to proving several important results in Steinberg algebras in the twisted setting, when R is a (discrete) field. In Section 6 we prove a twisted version of the Cuntz–Krieger uniqueness theorem for effective groupoids (Theorem 6.1), and we show that when R is a discrete field and G is effective, simplicity of AR(G,σ) is equivalent to minimality of G (Theorem 6.2). Finally, in Section 7, we show that twisted Steinberg algebras inherit a graded structure from the underlying groupoid, and we prove a graded uniqueness theorem for twisted Steinberg algebras (Theorem 7.2).

Section snippets

Preliminaries

In this section we introduce some notation, and we recall relevant background information on topological groupoids, continuous 2-cocycles, and twisted groupoid C*-algebras. Throughout this article, G will always be a locally compact Hausdorff topological groupoid with composable pairs G(2)G×G, range and source maps r,s:GG, and unit space G(0):=r(G)=s(G). We will refer to such groupoids as

. For all γG, we have r(γ)=γγ1 and s(γ)=γ1γ, where multiplication (or composition) of groupoid

Twisted Steinberg algebras arising from locally constant 2-cocycles

In this section we introduce the twisted Steinberg algebra AR(G,σ) over a discrete commutative unital ring R (or A(G,σ) when R=Cd) associated to an ample Hausdorff groupoid G and a continuous 2-cocycle σ:G(2)TR×. As an R-module, the twisted Steinberg algebra is identical to the untwisted version defined in Section 2. That is,AR(G,σ):=spanR{1B:GR:B is a compact open bisection of G}; we now emphasise that we are viewing R with the discrete topology.

Lemma 3.1

Let G be an ample Hausdorff groupoid, and let

Twisted Steinberg algebras arising from discrete twists

There is another (often more general) notion of a twisted groupoid C*-algebra which is constructed from a “twist” over the groupoid itself; that is, from a locally split groupoid extension of a Hausdorff étale groupoid G by G(0)×T. In this section, we define a discretised algebraic analogue of this twist and its associated twisted Steinberg algebra. The primary modification is to replace the topological group T with a discrete subgroup T of R×. Many of the results in Sections 4.1 and 4.2 have

Examples of twisted Steinberg algebras

In this section we discuss two important classes of examples of twisted Steinberg algebras: twisted group algebras and twisted Kumjian–Pask algebras.

A Cuntz–Krieger uniqueness theorem and simplicity of twisted Steinberg algebras of effective groupoids

In this section we extend the Cuntz–Krieger uniqueness theorem and a part of the simplicity characterisation for Steinberg algebras from [5] to the twisted Steinberg algebra setting. Throughout this section, we will assume that G is an effective ample Hausdorff groupoid, and that R=Fd is a field endowed with the discrete topology.

Theorem 6.1 Cuntz–Krieger uniqueness theorem

Let Fd be a discrete field, let G be an effective ample Hausdorff groupoid, and let σ:G(2)Fd× be a continuous 2-cocycle. Suppose that Q is a ring and that π:AFd(G,σ)Q

Gradings and a graded uniqueness theorem

In this section we describe the graded structure that twisted Steinberg algebras inherit from the underlying groupoid, and we prove a graded uniqueness theorem. The arguments are similar to those used in the untwisted setting (see [8]). Let Γ be a discrete group, and suppose that c:GΓ is a continuous groupoid homomorphism (or

). Then we call G a
, and we define Gγ:=c1(γ) for each γΓ. Since c is continuous and Γ is discrete, each Gγ is clopen. Since c is a homomorphism, we have

Acknowledgements

This research collaboration began as part of the project-oriented workshop “Women in Operator Algebras” (18w5168) in November 2018, which was funded and hosted by the Banff International Research Station. The attendance of the first-named author at this workshop was supported by an AustMS WIMSIG Cheryl E. Praeger Travel Award, and the attendance of the third-named author was supported by SFB 878 Groups, Geometry & Actions. The research was also funded by the Australian Research Council grant

References (36)

  • C. Bönicke

    K-theory and homotopies of twists on ample groupoids

    J. Noncommut. Geom.

    (2021)
  • J.H. Brown et al.

    Simplicity of algebras associated to étale groupoids

    Semigroup Forum

    (2014)
  • J.H. Brown et al.

    Decomposing the C*-algebras of groupoid extensions

    Proc. Am. Math. Soc.

    (2014)
  • K.S. Brown

    Cohomology of Groups

    (1982)
  • L.O. Clark et al.

    Uniqueness theorems for Steinberg algebras

    Algebr. Represent. Theory

    (2015)
  • L.O. Clark et al.

    A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras

    Forum Math.

    (2018)
  • L.O. Clark et al.

    A groupoid generalization of Leavitt path algebras

    Semigroup Forum

    (2014)
  • L.O. Clark et al.

    The equivalence relations of local homeomorphisms and Fell algebras

    N.Y. J. Math.

    (2013)
  • Cited by (13)

    • A uniqueness theorem for twisted groupoid C*-algebras

      2022, Journal of Functional Analysis
      Citation Excerpt :

      Renault's reconstruction theorem is of particular importance to the classification program for C*-algebras, given Li's recent article [32] showing that every simple classifiable C*-algebra has a Cartan subalgebra (and is therefore a twisted groupoid C*-algebra), and the work of Barlak and Li [7,8] describing the connections between the UCT problem and Cartan subalgebras in C*-algebras. The increasing interest in twisted groupoid C*-algebras (see, for instance, [3,6,10,13,16–18,25]) has also recently inspired the introduction of twisted Steinberg algebras, which are a purely algebraic analogue of twisted groupoid C*-algebras (see [4,5]). Examples of twisted groupoid C*-algebras include the twisted C*-algebras associated to higher-rank graphs introduced by Kumjian, Pask, and Sims [26–29], and the more general class of twisted C*-algebras associated to topological higher-rank graphs introduced in the author's PhD thesis [2].

    • TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

      2023, Journal of the Australian Mathematical Society
    View all citing articles on Scopus
    View full text