\(*\)-regular Leavitt path algebras of arbitrary graphs.
type: Item
instance of: scholarly article
title: \(*\)-regular Leavitt path algebras of arbitrary graphs.
zbMATH Open document ID: 1279.16003
author: Gonzalo Aranda Pino
author: Kulumani M. Rangaswamy
author: Lia Vaš
publication date: 2013-07-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.0379
review text: For a positive integer \(n\), an involution \(*\) on a ring \(R\) is said to be \(n\)-proper if for any \(x_1,\dots,x_n\in R\) one has that \(x_1x_1^*+\cdots+x_nx_n^*=0\) implies \(x_i=0\) for any \(i\). When \(n=1\) we say that the involution is proper. The involution \(*\) is positive-definite if it is \(n\)-proper for every \(n\). If \(K\) is a field with involution and \(E\) an arbitrary graph, the involution of \(K\) induces an involution of the Leavitt path algebra \(L_K(E)\). It is proved in this work that the involution in \(L_K(E)\) is proper if the involution in \(K\) is positive-definite. Furthermore the following three assertions are proved to be equivalent for a field \(K\) with involution: (i) the involution on \(K\) is positive-definite, (ii) The involution on \(L_K(E)\) is positive-definite for every graph \(E\); and (iii) The involution on \(L_K(E)\) is positive-definite for some graph \(E\). In section 4 the authors investigate the question of \(*\)-regularity of Leavitt path algebras. It is known that \(L_K(E)\) is (von Neumann) regular if the graph \(E\) is acyclic. It is proved in this section that for a ring \(R\) with local units, \(R\) is \(*\)-regular if and only if \(R\) is regular and \(*\) is proper. Then the characterization of \(*\)-regularity can be stated in the following terms: let \(K\) be a field with involution and \(E\) an arbitrary graph, the following assertions are equivalent: (i) \(L_K(E)\) is \(*\)-regular, (ii) \(L_K(E)\) is regular and proper, (iii) \(E\) is acyclic and \(K\) is \(n\)-proper for some finite \(n\) (more details on this \(n\) in the paper). As a consequence, it is proved that Handelman's conjecture (stating that every \(*\)-regular ring is unit-regular) holds for Leavitt path algebras. Moreover: its generalized version for rings with local units is also true for Leavitt path algebras over arbitrary graphs.
Mathematics Subject Classification ID: 16S88
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6188451
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: \(*\)-regular algebras
zbMATH Keywords: positive-definite involutions
zbMATH Keywords: Handelman conjecture
zbMATH Keywords: fields with involution
zbMATH Keywords: von Neumann regular algebras
reviewed by: Candido Martín González
MaRDI profile type: Publication
OpenAlex ID: W1998776649
arXiv ID: 1302.0379
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DOI: 10.1007/S10114-011-0106-8
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 6188451