A groupoid generalisation of Leavitt path algebras

Clark, Lisa Orloff; Farthing, Cynthia; Sims, Aidan; Tomforde, Mark

doi:10.1007/s00233-014-9594-z

Cited by 89 publications

(136 citation statements)

References 23 publications

(42 reference statements)

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“…Our proofs are similar in structure to the analogous proofs in section 5 of [4] where R = C. However in [4] the authors use the inner-product of C to build a function in the Steinberg algebra that is nonzero on G (0) . In the following key lemma, we produce the desired function in our more general setting.…”

Section: The Uniqueness Theoremsmentioning

confidence: 72%

“…These theorems generalise the analogous graph algebra uniqueness theorems and give conditions under which an R-algebra homomorphism π : A R (G) → A is injective. Complex Steinberg algebra versions of these theorems are given in [4] but the proofs rely on the standard inner product in C. We noticed that the analytic structure of C is not required and we simplify the arguments considerably.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness Theorems for Steinberg Algebras

Clark

Edie-Michell

2015

Algebr Represent Theor

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Section: The Uniqueness Theoremsmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Uniqueness Theorems for Steinberg Algebras

Clark

Edie-Michell

2015

Algebr Represent Theor

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“…When G is the graph groupoid G E of a directed graph E, the Steinberg algebra A R (G E ) is precisely the Leavitt path algebra L R (E), and the diagonal subalgebra D R (G E ) is precisely the diagonal subalgebra D R (E) of L R (E) ( [17,Remark 4.4…”

Section: Motivation and Historical Backgroundmentioning

confidence: 99%

“…Steinberg algebras, introduced in [30] and independently in [17], are algebraic analogues of groupoid C * -algebras. 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]).…”

Section: Introductionmentioning

confidence: 99%

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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“…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

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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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A groupoid generalisation of Leavitt path algebras

Cited by 89 publications

References 23 publications

Uniqueness Theorems for Steinberg Algebras

Uniqueness Theorems for Steinberg Algebras

Diagonal-preserving graded isomorphisms of Steinberg algebras

Cohn Path Algebras of Higher-Rank Graphs

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