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It is shown that there is a simple separable AF algebra A such that M(X 0 A) does not have weak (FN) and such that the generalized Berg-Weylvon Neumann Theorem does not hold for % 0 A. Consider two properties enjoyed by every selfadjoint element h of L(H) = M(Z) . (1) h can be approximated in norm by selfadjoint elements with finite spectrum. (2) There is a sequence of mutually orthogonal compact projections (e") and a bounded sequence of real numbers (in) such that h E' I Anen E X (the Weyl-von Neumann Theorem). A C*-algebra is said to be of real rank zero if (1) holds for every selfadjoint element h. The generalized Weyl-von Neumann Theorem for a C*-algebra A states that (2) holds for every selfadjoint element in the multiplier algebra M(A), where X is replaced by A (and the projections (en) belong to A and sum to the identity). These generalizations of (1) and (2) have attracted much attention in recent years. In particular, when A is c-unital and has real rank zero, it has been shown that M(A) has real rank zero if and only if the generalized Weyl-von Neumann Theorem holds for A [Linl, Zha]. Moreover, Lin has shown that if A is a c-inital AF algebra, then M(A) has real rank zero [Lin2]. As AF algebras are in many ways the simplest generalization of the algebra X, this result is very encouraging. Consider now analogues of properties (1) and (2) for normal operators. A C*-algebra is said to have property (FN) if every normal element can be approximated in norm by normal elements having finite spectrum. By the spectral theorem, L(H) has (FN). In more general multiplier algebras this definition is too strong. In a C*-algebra whose K1 group is nontrivial, an element x might fail to be approximated by elements with finite spectrum due to index obstructions corresponding to holes in the spectrum of x . Lin has defined weak (FN) to take such obstructions into account [Lin3, Definition 4.2]. Without recalling Received by the editors November 9, 1992. 1991 Mathematics Subject Classification. Primary 46L05. The first author was partially supported by National Science Foundation grant DMS-9007347. The second author was partially supported by National Science Foundation grant DMS-9102971. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

Approximation of normal elements in the multiplier algebra of an AF 𝐶*-algebra