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

David Handelman

1982
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Transactions of the American Mathematical Society
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Let A, C be approximately finite dimensional (AF) C* 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 Extz( K0(C),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
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... ed. 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). Our motivating theme is to consider extensions of C* algebras by other than the algebra of compact operators. Because AF algebras are describable in terms of partially ordered groups, they seem particularly suitable for this extension theory. As the ordered groups arising from simple AF algebras are fairly well understood, it turns out that one can solve completely the problem of classifying simple by simple AF algebras, when the finite trace space of the bottom AF algebra has finite dimensional dual. In the course of doing this, we establish formulae for the usual strong and weak Ext groups of AF algebras; our homological approach to this differs from the computational viewpoint of Pimsner and Popa [14, 15] . We consider short exact sequences ("extensions") of C* algebras, A -> B -> C, with B an AF algebra. There is a translation to extensions within a class of partially ordered abelian groups and distinguished subset, known as dimension groups with interval, via the functor K0. This translation is reversible (owing to a recent result of L. Brown that an extension of an AF algebra by an AF algebra is AF), so all C* extensions are represented as dimension group extensions. With the appropriate notion of equivalence (extending strong equivalence as defined in [3, p. 268], when A is the algebra of compact operators on a separable Hubert space), the equivalence classes admit a limited additive operation, often forming a disjoint union of groups. A single group results if, for example, both A and C are simple (with A unitless, but not necessarily stable). §1 deals with the appropriate definitions of extensions, dimension groups, equivalence, and the translation between AF algebras and dimension groups. Much of this is well known. In the second section, it is shown that if A is simple stable, and C is

doi:10.1090/s0002-9947-1982-0654850-0
fatcat:uoj4aafa6vbq5f7d3msoz6xreu