Uniqueness Theorems for Steinberg Algebras

Clark, Lisa Orloff; Edie-Michell, Cain

doi:10.1007/s10468-015-9522-2

Cited by 51 publications

(58 citation statements)

References 9 publications

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“…(13) The groupoids G E and G F are groupoid equivalent. (14) There are idempotents p E ∈ D R (E) and p F ∈ D R (F) such that p E is full in L R (E) and p F is full in L R (F), and a ring-isomorphism φ :…”

Section: 4mentioning

confidence: 99%

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Diagonal-preserving graded isomorphisms of Steinberg algebras

Carlsen

Rout

2018

Commun. Contemp. Math.

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ABSTRACT. 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 diagonalpreserving (graded) isomorphism between two (graded) Steinberg algebras. We apply this characterisation to groupoids of directed graphs in order to study diagonalpreserving (graded) isomorphisms of Leavitt path algebras and * -isomorphisms of graph C * -algebras.

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Section: 4mentioning

confidence: 99%

“…The class of Steinberg algebras includes for instance discrete inverse semigroup algebras (see for example [30]), Kumjian-Pask algebras (see for example [20]), and Leavitt path algebras (see for example [21]). Steinberg algebras have recently attracted a great deal of attention (see for instance [3,14,15,16,18,31,32]). …”

Section: Introductionmentioning

confidence: 99%

Diagonal-preserving graded isomorphisms of Steinberg algebras

Carlsen

Rout

2018

Commun. Contemp. Math.

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“…Steinberg algebras also support a Cuntz-Krieger Uniqueness Theorem and a Graded Uniqueness Theorem. These were first investigated in [19] and later improved in [24] and [64]. One can think of the Cuntz-Krieger Uniqueness Theorems as saying that a certain property of a graph, namely Condition (L), or a certain property of an ample groupoid, namely effectiveness, forces a homomorphism to be injective -provided it does not annihilate any scalar multiples of a local unit.…”

Section: Proofmentioning

confidence: 99%

“…The following result is an analogue of [2, Corollary 2.2.13], and it is just an alternative way of presenting some content from [24] and [64].…”

Section: Proofmentioning

confidence: 99%

The Groupoid Approach to Leavitt Path Algebras

Rigby

2020

Leavitt Path Algebras and Classical K-Theory

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When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph's boundary path groupoid.This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.

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“…are locally constant and have compact support (see [12,13,39]). Addition and scalar multiplication of A R (G) are defined pointwise, and convolution is given by…”

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confidence: 99%

Cohn Path Algebras of Higher-Rank Graphs

Clark

Pangalela

2016

Algebr Represent Theor

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Abstract. In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph Λ, there exists a higher-rank graph T Λ such that the Cohn path algebra of Λ is isomorphic to the Kumjian-Pask algebra of T Λ. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras. IntroductionLeavitt path algebras were introduced and studied in [3] and [9] as a generalisation the class of algebras studied by Leavitt in [25]. Leavitt path algebras are also the natural algebraic analogues of graph C * -algebras studied in [32]; a number of interesting result have been proven by translating between graph C * -algebras and the ring theoretic Leavitt path algebras.Cohn path algebras were introduced in [6,7] and generalise the algebras U 1,n studied by Cohn in [16]. The idea is to build an algebra out of the path space of a graph; addition and scalar multiplication are defined formally and multiplication of two paths is only nonzero when one can concatenate the paths. Cohn path algebras can also be obtained from Leavitt path algebras by omitting one of the Cuntz-Krieger relations. Hence every Leavitt path algebra can be viewed as a quotient of a Cohn path algebra. On the other hand, Abrams, Ara and Siles Molina show in [2] that for every graph E, there exists a graph T E (denoted E (X) in [2]) such that the Cohn path algebra of E is isomorphic to the Leavitt path algebra of T E.Although it has received less attention, the Cohn path algebra of a directed graph is the algebraic analogue of the C * -algebraic Toeplitz algebra of E as defined in [19]. In the C * -algebraic setting, Muhly and Tomforde in [27] and also Sims in [38] each show that for a graph E, the Toeplitz algebra of E is isomorphic to the graph C * -algebra of T E. Now we move into the setting of higher-rank graph algebras: In [24], Kumjian and Pask introduced a combinatorial model, called a higher-rank graph, in order to capture the essential features of the C * -algebras studied by Robertson and Steger in [35]. A higherrank graph, also called a k-graph, is a generalisation of the path category of a directed graph where the length of a 'path' λ in a k-graph is an element of N k . Kumjian and Pask studied the C * -algebras associated to row-finite higher-rank graphs with no sources. Raeburn, Sims and Yeend generalised Kumjian and Pask's construction by describing the class of C * -algebras associated to more general higher-rank graphs in [30,31].1991 Mathematics Subject Classification. 16S99 (Primary); 16S10 (Secondary).

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Uniqueness Theorems for Steinberg Algebras

Cited by 51 publications

References 9 publications

Diagonal-preserving graded isomorphisms of Steinberg algebras

Diagonal-preserving graded isomorphisms of Steinberg algebras

The Groupoid Approach to Leavitt Path Algebras

Cohn Path Algebras of Higher-Rank Graphs

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