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Regularity Conditions for Arbitrary Leavitt Path Algebras

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Abstract

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L K (E) is locally K-matricial; that is, L K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) L K (E) is von Neumann regular. (2) L K (E) is π-regular. (3) E is acyclic. (4) L K (E) is locally K-matricial. (5) L K (E) is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.

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  1. Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO, 80933, USA

    Gene Abrams & Kulumani M. Rangaswamy

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  1. Gene Abrams
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  2. Kulumani M. Rangaswamy
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Correspondence to Gene Abrams.

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Abrams, G., Rangaswamy, K.M. Regularity Conditions for Arbitrary Leavitt Path Algebras. Algebr Represent Theor 13, 319–334 (2010). https://doi.org/10.1007/s10468-008-9125-2

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  • DOI: https://doi.org/10.1007/s10468-008-9125-2

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