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The generalized Berg theorem and BDF-theorem

Let A be a separable simple AF-algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element x E M(A), i.e., ir(x) is normal (r: M(A) -+ M(A)/A is the quotient map), having the form y + a for some normal element y C M(A) and a E A. We also show that a normal element x E M(A) can be quasidiagonalized if and only if the Fredholm index ind(A x) = 0 for all A 0 sp(7r(x)). In the case that A is a simple C*-algebra of real rank zero, with stable rank one and with continuous scale, K1 (A) = 0, and Ko (A) has countable rank, we show that a normal element x E M(A) with zero Fredholm index can be written as