Approximation by normal elements with finite spectra in C*-algebras of real rank zero

We study the problem when a normal element in a C*-algebra of real rank zero can be approximated by normal elements with finite spectra. We show that all purely infinite simple C*-algebras, irrational rotation algebras and some types of C*-algebras of inductive limit of the form C(X) x M(n) of real rank zero have the property weak (FN), i.e., a normal element x can be approximated by normal elements with finite spectra if and only if Gamma(x) = 0 (lambda-x is an element of In v(0)(A) for all lambda is not an element of sp(x)). For general C*-algebras with real rank zero, we show that a normal element x with dim sp(x) less than or equal to 1 can be approximated by normal elements with finite spectra if and only if Gamma(x) = 0. One immediate application is that if A is a simple C*-algebra with real rank zero which is an inductive limit of C*-algebras Of form C(X(n)) x M(m(n)), where each X(n) is a compact subset of the plane, then A is an AF-algebra if and only if K-1 (A) = 0.