| Summary: | This work shows that the Weyl-von Neumann theorem for unitaries holds for $\sigma$-unital $AF$-algebras and their multiplier algebras. Lin studies $E(X,A)$, the quotient of $\mathrm{{{{\mathbf{{Ext}}}}}}^{{eu}}_s(C(X),A)$ by a special class of trivial extension, dubbed totally trivial extensions. This leads to a BDF-type classification for extensions of $C(X)$ by a $\sigma$-unital purely infinite simple $C^*$-algebra with trivial $K_1$-group. Lin also shows that, when $X$ is a compact subset of the plane, every extension of $C(X)$ by a finite matroid $C^*$-algebra is totally trivial. Classification of these extensions for nice spaces is given, as are some other versions of the Weyl-von Neumann-Berg theorem.
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