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