Algebras and Representation Theory

, Volume 13, Issue 3, pp 319–334

Regularity Conditions for Arbitrary Leavitt Path Algebras



We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally K-matricial; that is, LK(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) LK(E) is von Neumann regular. (2) LK(E) is π-regular. (3) E is acyclic. (4) LK(E) is locally K-matricial. (5) LK(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.


Leavitt path algebra Acyclic graph von Neumann regular algebra 

Mathematics Subject Classifications (2000)

Primary 16S99 16E50 Secondary 16W50 46L89 


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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Colorado at Colorado SpringsColorado SpringsUSA

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