Structure of Nonunital Purely Infinite Simple Rings

Gonzalez-Barroso, M. A.; Pardo, Enrique

doi:10.1080/00927870500387820

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…The concept of purely infinite simple ring was generalized to the setting of rings with local units in [3], and of nonunital (but σ-unital) rings in [30]. Concretely, the previous characterization was generalized to the context of rings with local units as follows.…”

Section: Purely Infinite Ringsmentioning

confidence: 99%

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Non-simple purely infinite rings

Pino,

Goodearl,

Perera

et al. 2008

Preprint

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In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-) unital K-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over K with many nonsimple Leavitt path algebras is purely infinite. G. A. P. and M. S. M. were supported in part by the Spanish MEC and Fondos FEDER through project MTM2007-60333, and jointly by the Junta de Andalucía and Fondos FEDER through projects FQM-336 and FQM-2467; F. P. was supported in part by the DGI MEC-FEDER through Project MTM2005-00934 and the

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Section: Purely Infinite Ringsmentioning

confidence: 99%

“…Concretely, the previous characterization was generalized to the context of rings with local units as follows. [30,Remark 2.7] and also [10,Corollary 5.4]. For some classes of non-simple (properly) purely infinite rings, see Sections 4 and 7.…”

Section: Purely Infinite Ringsmentioning

confidence: 99%

Non-simple purely infinite rings

Pino,

Goodearl,

Perera

et al. 2008

Preprint

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“…Inspired by the success of classifying purely infinite simple C * -algebras, Ara, Goodearl and Pardo extended the notion of purely infinite C * -algebras to unital rings, and studied their basic properties [4]. Purely infinite rings were studied further in [1,3,6,19]. In particular, [6] develops a general theory of purely infinite rings by presenting an algebraic parallel of the work in [21] on purely infinite C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

Dense subalgebras of purely infinite simple groupoidC*-algebras

Brown¹,

Clark²,

Huef³

2020

Proceedings of the Edinburgh Mathematical Society

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A simple Steinberg algebra associated to an ample Hausdorff groupoid G is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid C * -algebra C * r (G) is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate * -algebra B(G) constructed using corners 1 U C * r (G)1 U for all compact open subsets U of the unit space of the groupoid. We then show that if G is minimal and effective, then B(G) is algebraically properly infinite if and only if C * r (G) is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

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Non-simple purely infinite rings

Pino¹,

Goodearl²,

Perera³

et al. 2010

ajm

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In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-) unital $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many non-simple Leavitt path algebras is purely infinite.

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Structure of Nonunital Purely Infinite Simple Rings

Abstract: Abstract. In this note, we study the notion of purely infinite simple ring in the case of non-unital rings, and we obtain an analog to Zhang's Dichotomy for σ-unital purely infinite simple C*-algebras in the purely algebraic context.

Cited by 3 publications

References 9 publications

Non-simple purely infinite rings

Non-simple purely infinite rings

Dense subalgebras of purely infinite simple groupoidC*-algebras

Non-simple purely infinite rings

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