Invariance of the Cuntz splice

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.

doi:10.1007/s00208-017-1570-y

Cited by 7 publications

(3 citation statements)

References 25 publications

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“…The Cuntz splice and Pulelehua moves are complicated moves designed (by Cuntz in [Cun86] and the authors with Restorff and Sørensen in [ERRS]) to change the graph essentially without changing the K-theory of the graph C * -algebra. Proving that these moves do not change the graph C * -algebra either is a key step for establishing classification ([Rør95], [ERRS17], [ERRS]) and the stable isomorphisms thus obtained are not concrete and cannot be expected to preserve any additional structure.…”

Section: Refined Movesmentioning

confidence: 99%

Refined moves for structure-preserving isomorphism of graph C*-algebras

Eilers,

Ruiz

2019

Preprint

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We formalize eight different notions of isomorphism among (unital) graph C * -algebras, and initiate the study of which of these notions may be described geometrically as generated by moves. We propose a list of seven types of moves that we conjecture has the property that the collection of moves respecting one of six notions of isomorphism indeed generate that notion, in the sense that two graphs are equivalent in that sense if and only if one may transform one into another using only these kinds of moves.We carefully establish invariance properties of each move on our list, and prove a collection of generation results supporting our conjecture with an emphasis on the gauge simple case. In two of the six cases, we may prove the conjecture in full generality, and in two we can show it for all graphs defining gauge simple C * -algebras. In the two remaining cases we can show the conjecture for all graphs defining gauge simple C *algebras provided that they are either finite or have at most one vertex allowing a path back to itself.

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Section: Refined Movesmentioning

confidence: 99%

Refined moves for structure-preserving isomorphism of graph C*-algebras

Eilers,

Ruiz

2019

Preprint

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“…In Section 6, we show how to describe the isomorphisms these matrix operations induce on the (stabilized) graph C * -algebras and their reduced filtered K-theory. In [ERRS16c], it was proved that Cuntz splicing a graph gives a graph whose C * -algebra is stably isomorphic to the C * -algebra of the original graph (generalizing a result in [Res06]). As we are proving a strong classification result, we will need to know more about what this stable isomorphism induces on the reduced filtered K-theory.…”

Section: Toke Meier Carlsen Gunnar Restorff and Efren Ruizmentioning

confidence: 87%

Strong classification of purely infinite Cuntz-Krieger algebras

Carlsen

Restorff

Ruiz

2017

Trans. Amer. Math. Soc. Ser. B

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Abstract. In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a * -isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.

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“…In recent years, there has been an enormous amount of work, led by Eilers and his collaborators (see, for example, [2,3,4,5,6,7,17]) on determining which moves on finite directed graphs generate the equivalence relations determined by various types of isomorphism of the associated C * -algebras. One spectacular example of this is [3,Theorem 3.1]: if E and F are graphs with finitely many vertices, then the graph C * -algebras C * (E) and C * (F ) are stably isomorphic if and only if E can be transformed into F using a finite sequence of in-splittings, out-splittings, reductions, additions of sinks, Cuntz splices, Pulelehua moves, and the inverses of these moves.…”

Section: Introductionmentioning

confidence: 99%

Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

Brownlowe¹,

Laca²,

Robertson³

et al. 2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalised gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.Date: December 24, 2018. 1991 Mathematics Subject Classification. 46L05.

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Invariance of the Cuntz splice

Cited by 7 publications

References 25 publications

Refined moves for structure-preserving isomorphism of graph C*-algebras

Refined moves for structure-preserving isomorphism of graph C*-algebras

Strong classification of purely infinite Cuntz-Krieger algebras

Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

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