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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extensions for AF $ C\sp{\ast}$ algebras and dimension groups


Author: David Handelman
Journal: Trans. Amer. Math. Soc. 271 (1982), 537-573
MSC: Primary 46L05; Secondary 06F20, 16A56, 46M20
MathSciNet review: 654850
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Abstract: Let $ A$, $ C$ be approximately finite dimensional $ ({\text{AF)}}\,{C^{\ast}}$ algebras, with $ A$ nonunital and $ C$ unital; suppose that either (i) $ A$ is the algebra of compact operators, or (ii) both $ A$, $ C$ are simple. The classification of extensions of $ A$ by $ C$ is studied here, by means of Elliott's dimension groups. In case (i), the weak Ext group of $ C$ is shown to be $ {\operatorname{Ext} _{\mathbf{Z}}}({K_0}(C),\,{\mathbf{Z}})$, and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when $ A$ has only finitely many pure finite traces (and somewhat more generally).


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0654850-0
PII: S 0002-9947(1982)0654850-0
Article copyright: © Copyright 1982 American Mathematical Society