On cleanness of AW∗-algebras

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.