C*-Algebras with Weak (FN)

LetAbe a simpleC*-algebra of real rank zero and stable rank one. We show that, for any normal elementx∈Awithλ−x∈Inv0(Ã) for all λ∉sp(x)there is a sequence of normal elementsxn∈Awith finite spectra such thatx=limn→∞ xn.We show this for more generalC*-algebras. We also point out that the condition thatAis simple cannot be removed