TAILIEUCHUNG - Báo cáo toán học: "Dimension groups and finite difference equations "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Kích thước các nhóm và các phương trình khác biệt hữu hạn. | J. OPERATOR THEORY 2 1979 215-231 Copyright by INCREST 1979 DIMENSION GROUPS AND FINITE DIFFERENCE EQUATIONS EDWARD G. EFFROS and CHAO-LIANG SHEN INTRODUCTION A modified form of the classical Murray and von Neumann dimension theory for von Neumann algebras has begun to play an important role in c -algebra theory. Roughly speaking one begins by identifying unitarily equivalent projections in a c -algebra A and then in the matrix algebras M A . Letting N0 A be the equivalence classes the direct sura of projections provides an abelian semigroup operation on N0 A . One then produces an abelian group KU A from Ao 4 in essentially the same way that one constructs the integers z from the non-negative integers No . F A is the Grothendieck completion of 7V0 y4 . The dimension of a projection is defined to be its image in The set No determines the usual ordering of z . one has that m n if and only if m n e No. One might therefore expect that the image of Nq A should define something like a partial ordering of Ao 4 and in fact this should generally be the case for Grothendieck completions. Intuitively the ordering of K0 A should enable one to compare the sizes of dimensions. This idea was used by Elliott 7 who showed that the approximately finite AF c -algebras A are classified to within stable isomorphism by the corresponding ordered groups K0 A . We recall that an AF algebra is just the closure of an ascending sequence of finite dimension c -alge-bras. The classification of such algebras had earlier been studied by Glimm Dixmier and Bratteli 1 3 10 Elliott called the ordered groups that arise the countable dimension groups. The latter have recently been completely characterized by simple interpolation properties 4 and an extensive effort to classify these groups has begun see 8 18 5 6 4 Given a countable dimension group G it would be of some interest to have methods for finding specific AF c -algebras A with G Ka A . In ordered group theoretic terms this .

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