Contractible subgraphs and Morita equivalence of graph 𝐶*-algebras

Crisp, Tyrone; Gow, Daniel

doi:10.1090/s0002-9939-06-08216-5

Cited by 12 publications

(15 citation statements)

References 7 publications

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“…Several authors in [1], [4], [5] and [21] analyzed certain operations on quivers which preserve the (graded) Morita equivalence class of the associated Leavitt path algebras. This results are algebraic analogue of the analytic results that proved in [10], [12], [13], [16], [17] and [18]. Note that in the above papers the authors explored the connections between symbolic dynamics and Leavitt path algebras (graph C * -algebras).…”

Section: Introductionsupporting

confidence: 58%

Singular equivalence of finite dimensional algebras with radical square zero

Nasr-Isfahani¹

2016

Preprint

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We prove that the Crisp and Gow's quiver operation on a finite quiver Q produces a new quiver Q ′ with fewer vertices, such that the finite dimensional algebras kQ/J 2 and kQ ′ /J 2 are singularly equivalent. This operation is a general quiver operation which includes as specific examples some operations which arise naturally in symbolic dynamics (e.g., (elementary) strong shift equivalent, (in-out) splitting, source elimination, etc.).

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Section: Introductionsupporting

confidence: 58%

Singular equivalence of finite dimensional algebras with radical square zero

Nasr-Isfahani¹

2016

Preprint

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“…We begin by establishing a Morita equivalence property which is similar to an analogous property of graph C * -algebras. The C * -result, called Move (R), was shown in [18, Proposition 3.2]; it followed as a special case of the result [14,Theorem 3.1]. Move (R) applies only to very restricted configurations of vertices and edges in a graph.…”

Section: Collapsing At a Regular Vertex That Is Not The Base Of A Loopmentioning

confidence: 99%

Corners of Leavitt path algebras of finite graphs are Leavitt path algebras

Abrams

Nam

2019

Preprint

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We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra LK (E) of a finite graph E with coefficients in a field K as a direct sum of matrix rings over K, direct sum with a corner of the Leavitt path algebra LK (F ) of a graph F for which every regular vertex is the base of a loop. Moreover, in this case one may transform the graph E into the graph F via some step-by-step procedure, using the "source elimination" and "collapsing" processes. We use this to establish the main result of the article, that every nonzero corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. Indeed, we prove a more general result, to wit, that the endomorphism ring of any nonzero finitely generated projective LK (E)-module is isomorphic to the Leavitt path algebra of a graph explicitly constructed from E. Consequently, this yields in particular that every unital K-algebra which is Morita equivalent to a Leavitt path algebra is indeed isomorphic to a Leavitt path algebra.

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“…Instead, we find its quasi-inverse reduction in the final list [ERRS16] of moves on graphs which encode all Morita equivalences between graph C * -algebras. Indeed, reduction -rather than delay -was a central ingredient in [Sø13], and it is more easily recognized as a special case of the general result of [CG06].…”

Section: Reductionmentioning

confidence: 99%

Moves on $k$-graphs preserving Morita equivalence

Eckhardt¹,

Fieldhouse²,

Gent³

et al. 2020

Preprint

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We initiate the program of extending to higher-rank graphs (k-graphs) the geometric classification of directed graph C * -algebras, as completed in the 2016 paper of Eilers, Restorff, Ruiz, and Sørensen [ERRS16]. To be precise, we identify four "moves," or modifications, one can perform on a k-graph Λ, which leave invariant the Morita equivalence class of its C * -algebra C * (Λ). These moves -insplitting, delay, sink deletion, and reduction -are inspired by the moves for directed graphs described by Sørensen [Sø13] and Bates-Pask [BP04]. Because of this, our perspective on k-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a k-graph and its underlying directed graph.1 A graph E has finitely many vertices iff C * (E) is unital.

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Contractible subgraphs and Morita equivalence of graph 𝐶*-algebras

Cited by 12 publications

References 7 publications

Singular equivalence of finite dimensional algebras with radical square zero

Singular equivalence of finite dimensional algebras with radical square zero

Corners of Leavitt path algebras of finite graphs are Leavitt path algebras

Moves on $k$-graphs preserving Morita equivalence

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