On cleanness of $AW^*$-algebras
Volume 287 / 2026
Studia Mathematica 287 (2026), 57-79
MSC: Primary 47A65; Secondary 46L99
DOI: 10.4064/sm250417-23-9
Published online: 10 February 2026
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 Vaš. 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.
