We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the well-known characterizations when S is a field or a commutative ring. We also provide a criterion for the Steinberg algebra of the graph groupoid associated to an arbitrary graph to be congruence-simple. Motivated by a result of Clark and Sims, we show that, over the Boolean semifield, the natural homomorphism from the Leavitt path algebra to the Steinberg algebra is an isomorphism if and only if the associated graph is row-finite. Moreover, we establish the Reduction Theorem and Uniqueness Theorems for Leavitt path algebras of row-finite graphs over the Boolean semifield.
We investigate the algebra of an ample groupoid, introduced by Steinberg, over a semifield S. In particular, we obtain a complete characterization of congruence-simpleness for Steinberg algebras of second-countable ample groupoids, extending the well-known characterizations when S is a field. We apply our congruence-simplicity results to tight groupoids of inverse semigroup representations associated to self-similar graphs.
In this paper, we show that a unital simple Steinberg algebra is central, and a nonunital simple Steinberg algebra has zero center. We identify the fields $K$ and Hausdorff ample groupoids $\mathcal{G}$ for which the simple Steinberg algebra $A_K(\mathcal{G})$ yields a simple Lie algebra $[A_K(\mathcal{G}), A_K(\mathcal{G})]$. We apply the obtained results on simple Leavitt path algebras, simple Kumjian-Pask algebras and simple Exel-Pardo algebras to determine their associated Lie algebras are simple. In particular, we give easily computable criteria to determine which Lie algebras of the form…
Classic work of Pierce and Dauns-Hofmann shows that biregular rings are dual to simple ring bundles over Stone spaces. We extend this duality to Steinberg rings, a purely algebraic generalisation of Steinberg algebras, and ringoid bundles over ample groupoids. We base this largely on an even more general extension of Lawson's noncommutative Stone duality, specifically between Steinberg semigroups, a generalisation of Boolean inverse semigroups, and category bundles over ample groupoids.
Steinberg Algebras of Ample Semicategories and their Boolean-Cartan Restriction Semigroups
Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；Institute of Mathematics, Physics and Mechanics（数学、物理和力学研究所）；University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学与物理学院）
For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$, Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra $A_R(\mathcal{G})$ and the category of modules over the isotropy group algebra $R\mathcal{G}_x$. In this paper, we prove a graded version of these functors and related results for the graded Steinberg algebra of a graded ample groupoid. As an application, the spectral simple and graded simple modules over the Leavitt path algebra $L_K(E)$ are classified. In particular, we show that many of previously known s…
Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra and then, when $\mathcal{G}$ is graded by a cocycle, we study graded ideals in $A_R(\mathcal {G})$. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.
We give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid $G$, and certain families of ideals in the group algebras of isotropy groups in $G$. This generalises a known ideal correspondence theorem for Steinberg algebras of strongly effective groupoids. We use this to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.
We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra $L_K(E)$ of an arbitrary graph $E$ with coefficients in a field $K$ is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every $K$-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a $C…
Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{*}$-algebra of an abeli…
We prove that $A_R(G) \otimes_R A_R(H) \cong A_R(G \times H)$, if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R} \otimes L_{3,R}$ and $L_{2,R} \otimes L_{2,R}$. Indeed, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $*$-isomorphism of Steinberg algebras over the intege…
Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it a…
For an ample Hausdorff groupoid $G$, and the Steinberg algebra $A_R(G)$ with coefficients in the commutative ring $R$ with unit, we describe the centraliser of subalgebra $A_R(U)$ with $U$ an open closed invariant subset of unit space of $G$. In particular, we obtain that the algebra of the interior of the isotropy is indeed the centraliser of the diagonal subalgebra of Steinberg algebra. This will unify several results in the literature and the corresponding results for Leavitt path algebras follow.
We construct the full and reduced C*-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same C*-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, étale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every $*$-homomorphism from Connes' space of functions to $B(\mathcal{H})$ is automatically I-norm bounded. Previously, this was only known for Hausdorff groupoids.
In this paper, we characterize properly purely infinite Steinberg algebras $A_K(\mathcal{G})$ for strongly effective, ample Hausdorff groupoids $\mathcal{G}$. As an application, when $Λ$ is a strongly aperiodic $k$-graph, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian-Pask algebra $\text{KP}_K(Λ)$, which may be determined by the proper infiniteness of vertex idempotents. In particular, for unital cases, we give simple graph-theoretic criteria for the (proper) pure infiniteness of $\text{KP}_K(Λ)$. Furthermore, since the complex Steinbe…
We study the interplay between Steinberg algebras and partial skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorf…
We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use this to characterise when there is a diagonal-preserving (graded) isomorphism between two (graded) Steinberg algebras. We apply this characterisation to groupoids of directed graphs in order to study diagonal-preserving (graded) isomorphisms of Leavitt path algebras and graph $C^*$-algebras.
We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit.
Let $G$ and $H$ be Hausdorff ample groupoids and let $R$ be a commutative unital ring. We show that if $G$ and $H$ are equivalent in the sense of Muhly-Renault-Williams, then the associated Steinberg algebras of locally constant $R$-valued functions with compact support are Morita equivalent. We deduce that collapsing a ``collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path $R$-algebra, and therefore a number of graphical constructions which yield Morita equivalent $C^*$-algebras also yield Morita eq…
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 $R^\times$, we study the algebra $A_R(G,σ)$ 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…
Kaplansky introduced the notions of CCR and GCR $C^*$-algebras because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^*$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.
We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $κ: \mathcal G \to Γ$, to a discrete group $Γ$, such that $κ^{-1}(γ)κ^{-1}(δ) = κ^{-1}(γδ)$, for all $γ, δ\in Γ$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced by a $\mathcal G_ε$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. A…
Given a partial action $π$ of an inverse semigroup $S$ on a ring $\mathcal{A}$ one may construct its associated skew inverse semigroup ring $\mathcal{A} \rtimes_πS$. Our main result asserts that, when $\mathcal{A}$ is commutative, the ring $\mathcal{A} \rtimes_πS$ is simple if, and only if, $\mathcal{A}$ is a maximal commutative subring of $\mathcal{A} \rtimes_πS$ and $\mathcal{A}$ is $S$-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore,…
We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of tw…
Cartan inverse semigroups and twisted Steinberg algebras
University of Sydney（悉尼大学）；Victoria University of Wellington（惠灵顿维多利亚大学）；Queen’s University Belfast（贝尔法斯特女王大学）；MSU-Iligan Institute of Technology（棉兰老国立理工大学）
We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (…
A simple Steinberg algebra associated to an ample Hausdorff groupoid $G$ is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate $*$-algebra $B(G)$ constructed using corners $1_U C^*_r(G) 1_U$ for all compact open subsets $U$ of the unit space of the g…
Twisted étale groupoid algebras have been studied recently in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper, we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups and we show that they are classified by Lausch's second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists…
We show the reduced $C^*$-algebra of a graded ample groupoid is a strongly graded $C^*$-algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We apply this result to get a theorem about the Leavitt path algebra and $C^*$-algebra of an arbitrary graph.
Given an action $φ$ of of inverse semigroup $S$ on a ring $A$ (with domain of $φ(s)$ denoted by $D_{s^*}$) we show that if the ideals $D_e$, with $e$ an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.
In this article we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^*$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions - via the construction of a sectional algebra - are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; Semidirect products of bundles correspond to "naïve" crossed prod…
We unify various étale groupoid reconstruction theorems such as: 1) Kumjian-Renault's reconstruction from a groupoid C*-algebra. 2) Exel's reconstruction from an ample inverse semigroup. 3) Steinberg's reconstruction from a groupoid ring. 4) Choi-Gardella-Thiel's reconstruction from a groupoid L^p-algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation < on S. The groupoid of <-ultrafilters is then isomorphic to the original groupo…
