# $C^*$-algebra extensions of $C(X)$

### About this Title

**Hua Xin Lin**

Publication: Memoirs of the American Mathematical Society

Publication Year
1995: Volume 115, Number 550

ISBNs: 978-0-8218-2611-9 (print); 978-1-4704-0129-0 (online)

DOI: http://dx.doi.org/10.1090/memo/0550

MathSciNet review: 1257081

MSC: Primary 46L80; Secondary 19K33, 46M20, 47A58

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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.

Readership

Research mathematicians.

### Table of Contents

**Chapters**

- Introduction
- I. Totally trivial extensions
- II. The functor $E(\cdot , A)$
- III. BDF theory for $C^*$-algebras with real rank zero
- IV. Extensions by finite matroid algebras