Refinement monoids with weak comparability and applications to regular rings and $C*$-algebras
HTML articles powered by AMS MathViewer
- by P. Ara and E. Pardo
- Proc. Amer. Math. Soc. 124 (1996), 715-720
- DOI: https://doi.org/10.1090/S0002-9939-96-03059-6
- PDF | Request permission
Abstract:
We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O’Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and $C^*$-algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.References
- P. Ara, K.R. Goodearl, E. Pardo and D. Tyukavkin, $K$-Theoretically simple von Neumann regular rings, Journal of Algebra 174 (1995), 659–677.
- Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
- Bruce Blackadar, Comparison theory for simple $C^*$-algebras, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54. MR 996438
- Bruce Blackadar and David Handelman, Dimension functions and traces on $C^{\ast }$-algebras, J. Functional Analysis 45 (1982), no. 3, 297–340. MR 650185, DOI 10.1016/0022-1236(82)90009-X
- Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
- K. R. Goodearl, von Neumann regular rings, 2nd ed., Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. MR 1150975
- K. R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, 1986. MR 845783, DOI 10.1090/surv/020
- K. C. O’Meara, Simple regular rings satisfying weak comparability, J. Algebra 141 (1991), no. 1, 162–186. MR 1118322, DOI 10.1016/0021-8693(91)90210-Y
- E. Pardo, On the classification of simple Riesz groups, Preprint.
- E. Pardo, Metric completions of ordered groups and $K_0$ of exchange rings, Preprint.
- Friedrich Wehrung, Injective positively ordered monoids. I, II, J. Pure Appl. Algebra 83 (1992), no. 1, 43–82, 83–100. MR 1190444, DOI 10.1016/0022-4049(92)90104-N
- Ramesh V. Garimella, Prime ideals and the continuity of derivations on integral domains, Indian J. Pure Appl. Math. 22 (1991), no. 12, 1013–1017. MR 1139638
Bibliographic Information
- P. Ara
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra (Barcelona), Spain
- MR Author ID: 206418
- Email: para@mat.uab.es
- E. Pardo
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra (Barcelona), Spain; Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra (Barcelona), Spain
- MR Author ID: 345531
- ORCID: 0000-0002-1909-2895
- Email: epardo@mat.uab.es
- Received by editor(s): May 19, 1994
- Received by editor(s) in revised form: September 21, 1994
- Additional Notes: The research of the authors was supported by grants from the DGICYT (Spain).
- Communicated by: Ken Goodearl
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 715-720
- MSC (1991): Primary 16E20, 16E50, 46L80, 19K14, 06F20
- DOI: https://doi.org/10.1090/S0002-9939-96-03059-6
- MathSciNet review: 1301484
Dedicated: Dedicat al petit Guillem