Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T23:37:37.167Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Alfsen, E. M., Compact convex sets and boundary integrals (Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
2. Burgess, W. D. and Handelman, D. E., The N*-metric completion of regular rings, Math. Ann. 261 (1982), 235–254.Google Scholar
3. Effros, E. G., Handelman, D. E. and Shen, C. -L., Dimension groups and their affine representations, Amer. J. Math. 102 (1980), 385–407.Google Scholar
4. Ehrlich, G., Unit-regular rings, Portugal. Math. 27 (1968), 209–212.Google Scholar
5. Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29–44.Google Scholar
6. Goodearl, K. R., Simple regular rings and rank functions, Math. Ann. 214 (1975), 267–287.Google Scholar
7. Goodearl, K. R., Completions of regular rings, Math. Ann. 220 (1976), 229–252.Google Scholar
8. Goodearl, K. R., Algebraic representations of Choquet simplexes, J. Pure Applied Algebra 11 (1977), 111–130.Google Scholar
9. Goodearl, K. R., von Neumann regular rings (Pitman, London, 1979).Google Scholar
10. Goodearl, K. R., Notes on real and complex C*-algebras (Shiva, Nantwich, Cheshire, 1982).Google Scholar
11. Goodearl, K. R., Metrically complete regular rings, Trans. Amer. Math. Soc. 272 (1982), 275–310.Google Scholar
12. Goodearl, K. R. and Handelman, D. E., Rank functions and K0 of regular rings, J. Pure Applied Algebra 7 (1976), 195–216.Google Scholar
13. Goodearl, K. R. and Handelman, D. E., Metric completions of partially ordered abelian groups, Indiana Univ. Math. J. 29 (1980), 861–895.Google Scholar
14. Goodearl, K. R., Handelman, D. E. and Lawrence, J. W., Affine representations of Grothendieck groups and applications to Rickart C*-algebras and ℵ0-continuous regular rings, Memoirs Amer. Math. Soc. 234 (1980).Google Scholar
15. Handelman, D. E., Simple regular rings with a unique rank function, J. Algebra 42 (1976), 60–80.Google Scholar
16. Handelman, D. E., Representing rank complete continuous rings, Can. J. Math. 28 (1976), 1320–1331.Google Scholar
17. Handelman, D., Higgs, D. and Lawrence, J., Directed abelian groups, countably continuous rings, and Rickart C*-algebras, J. London Math. Soc. 21 (1980), 193–202.Google Scholar
18. Harpe, P. de la and Skandalis, G., Déterminant associé à une trace sur une algèbre de Banach, Ann. Inst. Fourier Grenoble 34 (1984), 241–260.Google Scholar
19. Kado, J., Unit-regular rings and simple selfinjective rings, Osaka J. Math. 18 (1981), 55–61.Google Scholar
20. Namioka, I. and Phelps, R. R., Tensor products of compact convex sets, Pacific J. Math. 31 (1969), 469–480.Google Scholar