⁎-Clean rings; some clean and almost clean Baer ⁎-rings and von Neumann algebras

Abstract

A ring is clean (resp. almost clean) if each of its elements is the sum of a unit (resp. regular element) and an idempotent. In this paper we define the analogous notion for ⁎-rings: a ⁎-ring is ⁎-clean (resp. almost ⁎-clean) if its every element is the sum of a unit (resp. regular element) and a projection. Although ⁎-clean is a stronger notion than clean, for some ⁎-rings we demonstrate that it is more natural to use. The theorem on cleanness of unit-regular rings from Camillo and Khurana (2001) [4] is modified for ⁎-cleanness of ⁎-regular rings that are abelian (or reduced or Armendariz). Using this result, it is shown that all finite, type I Baer ⁎-rings that satisfy certain axioms (considered in Berberian (1972) [2] and Vaš (2005) [26]) are almost ⁎-clean. In particular, we obtain that all finite type I AW ⁎ -algebras (thus all finite type I von Neumann algebras as well) are almost ⁎-clean. We also prove that for a Baer ⁎-ring satisfying the same axioms, the following properties are equivalent: regular, unit-regular, left (right) morphic and left (right) quasi-morphic. If such a ring is finite and type I, it is ⁎-clean. Finally, we present some examples related to group von Neumann algebras and list some open problems.