Graded Steinberg algebras and their representations

Ara, Pere; Hazrat, Roozbeh; Li, Huanhuan; Sims, Aidan

doi:10.2140/ant.2018.12.131

Cited by 56 publications

(67 citation statements)

References 46 publications

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“…We also obtain several other results about (hermitian) graded K-theory of ( * -) algebras in general and Leavitt path algebras in particular which we think are of independent interest. Building upon work of Ara, Hazrat, Li and Sims in [2] and Preusser in [15], we show in Theorems 3.1.8 and 3.2.6 that if R is a ( * -) ring, graded over a group G, and having (self-adjoint) graded local units, then the (hermitian) graded K-theory of R is the (hermitian) K-theory of the crossed product (1.5)…”

Section: Introductionmentioning

confidence: 82%

“…The rest of this article is organized as follows. In Section 2 we recall basic definitions, notations and properties for algebras equipped with an action of, or a grading over a group G. Section 3 contains the proof of the identities (1.5) in Theorems 3.1.8 and 3.2.6; the basic idea is to use the category isomorphism between graded R-modules and G ⋉ R-modules due to Ara, Hazrat, Li and Sims [2] and the fact that the latter preserves finite generation, proved in Preusser's article [15], and to check that the category isomorphism intertwines the relevant duality functors. As an application we also establish Theorem 4.3, which is a hermitian variant of Dade's theorem.…”

Section: Introductionmentioning

confidence: 99%

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Graded $K$-theory and Leavitt path algebras

Arnone¹,

Cortiñas²

2022

Preprint

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Let G be a group and ℓ a commutative unital * -ring with an element λ ∈ ℓ such that λ + λ * = 1. We introduce variants of hermitian bivariant K-theory for * -algebras equipped with a G-action or a G-grading. For any graph E with finitely many vertices and any weight function ω :Ggr is obtained, describing L(E) as a cone of a matrix with coefficients in Z[G] associated to the incidence matrix of E and the weight ω. In the particular case of the standard Z-grading, and under mild assumptions on ℓ, we show that the isomorphism class of L(E) in kk hZgr is determined by the graded Bowen-Franks module of E. We also obtain results for the graded and hermitian graded K-theory of * -algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence relating graded and ungraded K-theory.Again the same formulas hold for homotopy algebraic and homotopy hermitian K-theory without any unitality assumptions on R. We also consider gradings on L(E) with values on a group G for an arbitrary graph E.

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Graded $K$-theory and Leavitt path algebras

Arnone¹,

Cortiñas²

2022

Preprint

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“…By [15,Theorem 4.12], E is a no-exit graph. Conversely, if E is no-exit, then L K (E) is cancellable by [5,Lemma 5.5]. So, C(P ) holds for every finitely generated projective L K (E)-module P which implies that IC(P ) holds for every such module P.…”

Section: 3mentioning

confidence: 97%

Graded Cancellation Properties of Graded Rings and Graded Unit-regular Leavitt Path Algebras

Vaš

2020

Algebr Represent Theor

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We raise the following general question regarding a ring graded by a group: "If P is a ring-theoretic property, how does one define the graded version P gr of the property P in a meaningful way?". Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If P is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first half of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness.In the second half of the paper, we turn to Leavitt path algebras. For Leavitt path algebras of finite graphs, we characterize graded unit-regularity and other cancellation properties in terms of the graph properties. Then we provide a complete description of graded matrix algebras over a trivially graded field which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to another Leavitt path algebra. If R is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which R can be realized as a Leavitt path algebra.

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“…The grading of a Leavitt path algebra is also fundamental in the understanding of its algebraic structure, in particular its ideal structure, see [2,26] for example.…”

Section: Introductionmentioning

confidence: 99%

Properties of the gradings on ultragraph algebras via the underlying combinatorics

Royer¹

2021

Preprint

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There are two established gradings on Leavitt path algebras associated with ultragraphs, namely the grading by the integers group and the grading by the free group on the edges. In this paper, we characterize properties of these gradings in terms of the underlying combinatorial properties of the ultragraphs. More precisely, we characterize when the gradings are strong or epsilon-strong. The results regarding the free group on the edges are new also in the context of Leavitt path algebras of graphs. Finally, we also describe the relation between the strongness of the integer grading on an ultragraph Leavitt path algebra and the saturation of the gauge action associated with the corresponding ultragraph C*-algebra.

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Graded Steinberg algebras and their representations

Cited by 56 publications

References 46 publications

Graded $K$-theory and Leavitt path algebras

Graded $K$-theory and Leavitt path algebras

Graded Cancellation Properties of Graded Rings and Graded Unit-regular Leavitt Path Algebras

Properties of the gradings on ultragraph algebras via the underlying combinatorics

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