Tensor Products of Dimension Groups and <i>K</i>₀ of Unit-Regular Rings

Abstract

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 ℵ 1 , 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].