On cleanness of AW*-algebras
Lu Cui, Minghui Ma
TL;DR
This work addresses cleanness in $AW^*$-algebras by extending the notions of clean and almost $*$-clean rings to the operator-algebra setting. It develops a toolkit based on left/right projections, spectral projections, Kaplansky-type formulas, and Halmos two-projections technology to study invertibility and decompositions. The main results show that all finite $AW^*$-algebras are clean with an explicit bound on the inverse norm, and that finiteness characterizes almost $*$-cleanness. Additionally, every countably decomposable infinite $AW^*$-factor is clean, advancing understanding of cleanness in infinite, noncommutative algebras and highlighting open questions about the cleanness of all infinite $AW^*$-algebras.
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.