Regularity conditions for arbitrary Leavitt path algebras

2008/06/23 by G. Abrams, Gene Abrams, Abrams, G. +3 · 2 citations
Computer Science · Mathematics · #16E50 #16S99 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #math.OA #math.RA #msc:16E50 #msc:16S99

paper · pdf · doi:10.48550/arxiv.0806.3743

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

openalex publication_date 2008/06/23 · arxiv created 2008/10/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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. 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.

Cited by

• The Local-Global Principle in Leavitt Path Algebras
• Extensions of locally matricial and locally semisimple algebras

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