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Extensions of dimension groups and AF C * -algebras.

Extensions of dimension groups and AF C * -algebras. Introduction The major purpose of this paper is to develop a method for classifying all (essential) unital C*-algebra extensions of an arbitrary AF (i.e., approximately finitedimensional) C*-algebra A by an arbitrary unital AF C*-algebra C. (This extends and completes earlier work of Handelman and the author in [10] and [9].) In particular, some existence theorems follow; e.g., if A is nonzero and has no nonzero unital homomorphic images, then (essential, unital) extensions of A by C always exist. Alternately put, this is an embedding theorem: if A is nonzero and has no nonzero unital homomorphic images, then every (unital) AF C*-algebra embeds (unitally) in the multiplier quotient Ji(A)/A. Since AF C*-algebras are determined up to isomorphism by their ordered Grothendieck groups (i.e., K0), the problem of classifying extensions of these algebras can be translated without loss of Information into a classification problem for extensions of a dimension group G± by a dimension group G2 . The latter problem is solved in the following manner: dimension group extensions of Gt by G2 are completely classified in terms of two pieces of data -- group homomorphisms from G2 to (Gi ® Q)/G1 together with monoid homomorphisms from G2 to a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik (Crelle's Journal) de Gruyter

Extensions of dimension groups and AF C * -algebras.

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0075-4102
eISSN
1435-5345
DOI
10.1515/crll.1990.412.150
Publisher site
See Article on Publisher Site

Abstract

Introduction The major purpose of this paper is to develop a method for classifying all (essential) unital C*-algebra extensions of an arbitrary AF (i.e., approximately finitedimensional) C*-algebra A by an arbitrary unital AF C*-algebra C. (This extends and completes earlier work of Handelman and the author in [10] and [9].) In particular, some existence theorems follow; e.g., if A is nonzero and has no nonzero unital homomorphic images, then (essential, unital) extensions of A by C always exist. Alternately put, this is an embedding theorem: if A is nonzero and has no nonzero unital homomorphic images, then every (unital) AF C*-algebra embeds (unitally) in the multiplier quotient Ji(A)/A. Since AF C*-algebras are determined up to isomorphism by their ordered Grothendieck groups (i.e., K0), the problem of classifying extensions of these algebras can be translated without loss of Information into a classification problem for extensions of a dimension group G± by a dimension group G2 . The latter problem is solved in the following manner: dimension group extensions of Gt by G2 are completely classified in terms of two pieces of data -- group homomorphisms from G2 to (Gi ® Q)/G1 together with monoid homomorphisms from G2 to a

Journal

Journal für die reine und angewandte Mathematik (Crelle's Journal)de Gruyter

Published: Jan 1, 1990

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