Partially Ordered Abelian Groups with Interpolation
About this Title
Kenneth R. Goodearl, University of California, Santa Barbara, CA
Publication: Mathematical Surveys and Monographs
Publication Year 1986: Volume 20
ISBNs: 978-0-8218-4980-4 (print); 978-1-4704-1247-0 (online)
MathSciNet review: MR845783
MSC: Primary 06F20; Secondary 16A54, 18F25, 19K14, 46A55, 46L80
A branch of ordered algebraic structures has grown, motivated by $K$-theoretic applications and mainly concerned with partially ordered abelian groups satisfying the Riesz interpolation property. This monograph is the first source in which the algebraic and analytic aspects of these interpolation groups have been integrated into a coherent framework for general reference. The author provides a solid foundation in the structure theory of interpolation groups and dimension groups (directed unperforated interpolation groups), with applications to ordered $K$-theory particularly in mind.
Although interpolation groups are defined as purely algebraic structures, their development has been strongly influenced by functional analysis. This cross-cultural development has left interpolation groups somewhat estranged from both the algebraists, who may feel intimidated by compact convex sets, and the functional analysts, who may feel handicapped by the lack of scalars. This book, requiring only standard first-year graduate courses in algebra and functional analysis, aims to make the subject accessible to readers from both disciplines.
High points of the development include the following: characterization of dimension groups as direct limits of finite products of copies of the integers; the double-dual representation of an interpolation group with order-unit via affine continuous real-valued functions on its state space; the structure of dimension groups complete with respect to the order-unit norm, as well as monotone sigma-complete dimension groups and dimension groups with countably infinite interpolation; and an introduction to the problem of classifying extensions of one dimension group by another. The book also includes a development of portions of the theory of compact convex sets and Choquet simplices, and an expository discussion of various applications of interpolation group theory to rings and $C^*$-algebras via ordered $K_0$. A discussion of some open problems in interpolation groups and dimension groups concludes the book.
Of interest, of course, to researchers in ordered algebraic structures, the book will also be a valuable source for researchers seeking a background in interpolation groups and dimension groups for applications to such subjects as rings, operator algebras, topological Markov chains, positive polynomials, compact group actions, or other areas where ordered Grothendieck groups might be useful.
Table of Contents
- 1. Basic notions
- 2. Interpolation
- 3. Dimension groups
- 4. States
- 5. Compact convex sets
- 6. State spaces
- 7. Representation by affine continuous functions
- 8. General comparability
- 9. Dedekind $\sigma $-completeness
- 10. Choquet simplices
- 11. Affine continuous functions on Choquet simplices
- 12. Metric completions
- 13. Affine continuous functions on state spaces
- 14. Simple dimension groups
- 15. Norm-completeness
- 16. Countable interpolation and monotone $\sigma $-completeness
- 17. Extensions of dimension groups