Purely infinite simple Leavitt path algebras.
type: Item
instance of: scholarly article
title: Purely infinite simple Leavitt path algebras.
zbMATH Open document ID: 1137.16028
author: Gonzalo Aranda Pino
author: Gene Abrams
published in: Journal of Pure and Applied Algebra
publication date: 2006-11-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0509496
review text: Leavitt path algebras are the algebraic version of Cuntz-Krieger graph \(C^*\)-algebras, a class of algebras intensively investigated by analysts for more than two decades. These algebras were introduced by \textit{G. Abrams} and \textit{G. Aranda Pino} [J. Algebra 293, No. 2, 319-334 (2005; Zbl 1119.16011)]. Abrams and Aranda Pino provided a generalization of Leavitt algebras \(L(1,n)\), of type \((1,n)\), which were introduced by \textit{W. G. Leavitt} in 1962 [in Trans. Am. Math. Soc. 103, 113-130 (1962; Zbl 0112.02701)] in order to give examples of algebras not satisfying the IBN (invariant basis number) property. An idempotent \(e\) in a ring \(R\) is said to be infinite if \(eR\) is isomorphic, as a right \(R\)-module, to a proper direct summand of itself. The ring \(R\) is called purely infinite if each of its nonzero right ideals contains an infinite idempotent. This notion was introduced in the algebraic setting by \textit{P. Ara, K. R. Goodearl} and \textit{E. Pardo} [K-Theory 26, No. 1, 69-100 (2002; Zbl 1012.16013)]. In this paper, the authors establish necessary and sufficient conditions on a row-finite graph \(E\) so that the associated Leavitt path \(K\)-algebra \(L_K(E)\) (for \(K\) a field) is purely infinite and simple. These are: (i) The only hereditary and saturated subsets of \(E^0\), the set of vertices of the graph, are the trivial ones (\(E^0\) and \(\emptyset\)), (ii) every cycle in \(E\) must have an exit, and (iii) every vertex connects to a cycle. The analytic counterpart of this result can be found in [\textit{T. Bates} et al., New York J. Math. 6, 307-324 (2000; Zbl 0976.46041)]. Although the algebraic result is the same in shape, the methods used differ from the analytic ones.
reviewed by: Pere Ara
Mathematics Subject Classification ID: 16S88
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 5073070
zbMATH Keywords: purely infinite simple Leavitt path algebras
zbMATH Keywords: idempotents
zbMATH Keywords: Cuntz-Krieger graph \(C^*\)-algebras
zbMATH Keywords: purely infinite rings
zbMATH Keywords: row-finite graphs
zbMATH Keywords: rings with local units
MaRDI profile type: Publication
OpenAlex ID: W2167935251
arXiv ID: math/0509496
cites work: The Leavitt path algebra of a graph.
cites work: Morita equivalence for rings without identity
cites work: The exchange property for purely infinite simple rings
cites work: Fractional skew monoid rings.
cites work: \(K_0\) of purely infinite simple regular rings
cites work: Nonstable \(K\)-theory for graph algebras.
cites work: The \(C^*\)-algebras of row-finite graphs
cites work: Cuntz-Krieger algebras of directed graphs
cites work: The Module Type of a Ring
cites work: The module type of homomorphic images
cites work: A classification theorem for nuclear purely infinite simple \(C^*\)-algebras
DOI: 10.1016/J.JPAA.2005.10.010
description: scientific article