2025/06/10 by Kulumani M. Rangaswamy, Rangaswamy, Kulumani M., Ashish K. Srivastava +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2506.08305
openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the graded version of Naimark's problem for Leavitt path algebras considering them as ℤ-graded algebras. Several characterizations are obtained of a Leavitt path algebra L of an arbitrary graph E over a field \mathbbK over which any two graded-simple modules are graded isomorphic. Such a Leavitt path algebra L is shown to be graded isomorphic to the algebra of graded infinite matrices having at most finitely many non-zero entries from the ring R where R=\mathbbK or R=\mathbbK[x,x-1]. Equivalently, L is a graded-simple ring which is graded-semisimple, that is, L is a graded direct sum of graded-isomorphic graded-simple left L-modules. Graphically, the graph E is shown to be row-finite, downward directed and the vertex set E0 is the hereditary saturated closure of a single vertex v which is either a line point or lies on a cycle without exits. We also characterize Leavitt path algebras possessing at most countably many isomorphism classes of graded-simple left modules. Examples are constructed illustrating these results.