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The generalized Berg theorem and BDF-theorem

Authors:	Huaxin Lin

Institution:	Department of Mathematics, East China Normal University, Shanghai 20062, China

Abstract:	Let be a separable simple -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 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 is a simple -algebra of real rank zero, with stable rank one and with continuous scale, and has countable rank, we show that a normal element $x\in M(A)$ with zero Fredholm index can be written as $\begin{equation}x=\sum _{n=1}^{\infty }\lambda _n(e_n-e_{n-1})+a, \end{equation}$ where $\{e_n\}$ is an (increasing) approximate identity for consisting of projections, $\{\lambda _n\}$ is a bounded sequence of numbers and $a\in A$ with $\\|a\\|<\epsilon$ for any given $\epsilon >0.$

Keywords:	The Berg theorem the BDF-theorem weak (FN) $C^*$-algebra with real rank zero

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