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 $x\in M(A)$, i.e., $\pi (x)$ is normal ($\pi : M(A)\to M(A)/A$ is the quotient map), having the form $y+a$ for some normal element $y\in M(A)$ and $a\in A.$ We also show that a normal element $x\in M(A)$ can be quasi-diagonalized if and only if the Fredholm index $ind(\lambda -x)=0$ for all $\lambda \not\in sp(\pi (x)).$ In the case that $A$ is a simple $C^*$-algebra of real rank zero, with stable rank one and with continuous scale, $K_1(A)=0,$ and $K_0(A)$ has countable rank, we show that a normal element $x\in M(A)$ with zero Fredholm index can be written as

where $\{e_n\}$ is an (increasing) approximate identity for $A$ consisting of projections, $\{\lambda _n\}$ is a bounded sequence of numbers and $a\in A$ with $\|a\|<\epsilon$ for any given $\epsilon >0.$