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Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

Published online by Cambridge University Press:  20 November 2018

K. R. Goodearl  and
D. E. Handelman
K. R. Goodearl
Affiliation:
University of Utah, Salt Lake City, Utah
D. E. Handelman
Affiliation:
University of Ottawa, Ottawa, Ontario
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We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 3 , 01 June 1986 , pp. 633 - 658
Copyright
Copyright © Canadian Mathematical Society 1986

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