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.