On cleanness of von Neumann algebras
Lu Cui, Linzhe Huang, Wenming Wu, Wei Yuan, Hanbin Zhang
TL;DR
This paper characterizes strongly clean von Neumann algebras and proves that all finite von Neumann algebras and separable infinite factors are clean, advancing understanding of their algebraic structure.
Contribution
It provides a complete characterization of strongly clean von Neumann algebras and establishes cleanness for broad classes of these algebras.
Findings
Finite von Neumann algebras are clean.
Separable infinite factors are clean.
Strongly clean von Neumann algebras are characterized.
Abstract
A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.
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Taxonomy
TopicsRings, Modules, and Algebras · Algebraic structures and combinatorial models · Advanced Algebra and Logic