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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The generalized Berg theorem and BDF-theorem

Author: Huaxin Lin
Journal: Trans. Amer. Math. Soc. 349 (1997), 529-545
MSC (1991): Primary 46L05
MathSciNet review: 1401777
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Abstract: 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

\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 $A$ consisting of projections, $\{\lambda _n\}$ is a bounded sequence of numbers and $a\in A$ with $\|a\|<\epsilon $ for any given $\epsilon >0.$

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Additional Information

Huaxin Lin
Affiliation: Department of Mathematics, East China Normal University, Shanghai 20062, China
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Keywords: The Berg theorem, the BDF-theorem, weak (FN), $C^*$-algebra with real rank zero
Received by editor(s): July 27, 1993
Article copyright: © Copyright 1997 American Mathematical Society