Tensor Products of Dimension Groups and K 0 of Unit-Regular Rings

Goodearl, K. R. ; Handelman, D. E.

Canadian journal of mathematics, Tome 38 (1986) no. 3, pp. 633-658

Résumé

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K 0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K 0 of a unit-regular ring or even as K 0 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 K 0 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].

Détail

DOI : 10.4153/CJM-1986-032-0

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