On cleanness of AW*-algebras

2025/04/18 by Cui, Lu, Ma, Minghui
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2504.13470

Abstract

A ring is called clean if every element is the sum of an invertible element and an idempotent. This paper investigates the cleanness of AW*-algebras. We prove that all finite AW*-algebras are clean, affirmatively solving a question posed by Vas. We also prove that all countably decomposable infinite AW*-factors are clean. A *-ring is called almost *-clean if every element can be expressed as the sum of a non-zero-divisor and a projection. We show that an AW*-algebra is almost *-clean if and only if it is finite.

Citations

• On cleanness of von Neumann algebras
• Classes of almost clean rings
• *-Clean Rings; Some Clean and Almost Clean Baer *-rings and von Neumann Algebras
• Continuous modules are clean
• Spectral and polar decomposition in AW*-algebras
• Perspectivity and cancellation in regular rings
• Lifting idempotents and exchange rings
• Projections in Banach Algebras

Related

• On cleanness of von Neumann algebras
• On *-clean group rings over finite fields
• Two questions of L. Va\rm \checks on *-clean rings
• Very clean matrices over local rings
• r-clean rings