Regularity Conditions for Arbitrary Leavitt Path Algebras

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 pi-regular. (3) E is acyclic. (4) L (K) (E) is locally K-matricial. (5) L (K) (E) is strongly pi-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.