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$$C^*$$-Algebra Extensions of $$C(X)$$
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Memoirs of the American Mathematical Society
1995; 89 pp; softcover
Volume: 115
ISBN-10: 0-8218-2611-5
ISBN-13: 978-0-8218-2611-9
List Price: US$39 Individual Members: US$23.40
Institutional Members: US\$31.20
Order Code: MEMO/115/550

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.

• The functor $$E(\cdot \,, A)$$
• BDF theory for $$C^*$$-algebras with real rank zero