Extensions for AF 𝐶* algebras and dimension groups

Handelman, David

doi:10.1090/S0002-9947-1982-0654850-0

Extensions for AF $C^{\ast }$ algebras and dimension groups
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Trans. Amer. Math. Soc. 271 (1982), 537-573 Request permission

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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