Simple Cuntz-Pimsner rings.
type: Item
instance of: scholarly article
title: Simple Cuntz-Pimsner rings.
zbMATH Open document ID: 1273.16006
author: Toke Meier Carlsen
author: Eduard Ortega
author: Enrique Pardo Espino
published in: Journal of Algebra
publication date: 2013-05-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.6923
review text: Relative Cuntz-Pimsner rings are algebraic analogues of relative Cuntz-Pimsner \(C^*\)-algebras. This kind of rings generalizes several algebraic structures among which we can mention Leavitt path algebras. As in the case of Leavitt path algebras, relative Cuntz-Pimsner rings come equipped with a \(\mathbb Z\)-grading. In a paper by the two first authors in 2011, a complete description of all graded ideals of an arbitraty Cuntz-Pimsner ring is given. In the present paper, necessary and sufficient conditions are given to ensure that a nonzero ideal in a relative Cuntz-Pimsner ring contains a nonzero-graded ideal. Then, simplicity of relative Cuntz-Pimsner rings is characterized. Also necessary and sufficient conditions are given to determine when every ideal in a relative Cuntz-Pimsner ring is graded. The authors also introduce the natural analogues of condition (L) and condition (K) in relative Cuntz-Pimsner rings, so as to get results extending the ones for the corresponding classes of Leavitt path algebras. Finally the paper deals with a uniqueness result for the Toeplitz algebra of a directed graph and with characterizations of simple crossed products of a ring by a single automorphism and simple fractional skew monoid rings of a single corner isomorphism.
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 6166664
zbMATH Keywords: Cuntz-Pimsner rings
zbMATH Keywords: simplicity
zbMATH Keywords: invariant cycles
zbMATH Keywords: condition (K)
zbMATH Keywords: condition (L)
zbMATH Keywords: Cuntz-Krieger uniqueness
zbMATH Keywords: Toeplitz rings
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: crossed products
zbMATH Keywords: fractional skew monoid rings
zbMATH Keywords: graded ideals
reviewed by: Candido Martín González
MaRDI profile type: Publication
OpenAlex ID: W2036396741
arXiv ID: 1110.6923
cites work: The Leavitt path algebra of a graph.
cites work: Fractional skew monoid rings.
cites work: Multipliers of von neumann regular rings
cites work: Algebraic Cuntz-Pimsner rings
cites work: The Toeplitz algebra of a Hilbert bimodule
cites work: On \(C^*\)-algebras associated with \(C^*\)-correspondences
cites work: A class of ${C^*}$-algebras generalizing both graph algebras and homeomorphism ${C^*}$-algebras III, ideal structures
cites work: Ideal structure of \(C^*\)-algebras associated with \(C^*\)-correspondences
cites work: Modules without Invariant Basis Number
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Tensor algebras over \(C^*\)-correspondences: Representations, dilations, and \(C^*\)-envelopes
cites work: Simple group graded rings and maximal commutativity
cites work: Q3995022
cites work: Q4335109
cites work: Q4681774
cites work: Relative Cuntz-Krieger algebras of finitely aligned higher-rank graphs
cites work: Simplicity of ultragraph algebras
cites work: Uniqueness theorems and ideal structure for Leavitt path algebras
DOI: 10.1016/J.JALGEBRA.2012.08.005
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description: scientific article