Regularity conditions for arbitrary Leavitt path algebras

Details Regularity conditions for arbitrary Leavitt path algebras Regularity conditions for arbitrary Leavitt path algebras

2008-06-23

arxiv.org/abs/0806.3743v2

Mathematics

Rings and Algebras

15 pages, accepted version July 2008 to appear Algebras and Representation Theory

Scientific paper

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 $K$. 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.

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