**Mathematical Surveys and Monographs**

Volume: 20;
1986;
336 pp;
Softcover

MSC: Primary 06;
Secondary 16; 18; 19; 46

Print ISBN: 978-0-8218-4980-4

Product Code: SURV/20.S

List Price: $103.00

Individual Member Price: $82.40

**Electronic ISBN: 978-1-4704-1247-0
Product Code: SURV/20.S.E**

List Price: $103.00

Individual Member Price: $82.40

# Partially Ordered Abelian Groups with Interpolation

Share this page
*Kenneth R. Goodearl*

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

# Table of Contents

## Partially Ordered Abelian Groups with Interpolation

- CONTENTS v6 free
- PREFACE ix10 free
- PROLOGUE: PARTIALLY ORDERED GROTHENDIECK GROUPS xiii14 free
- NOTATIONAL CONVENTIONS xxi22 free
- 1. BASIC NOTIONS 124 free
- 2. INTERPOLATION 2245
- 3. DIMENSION GROUPS 4467
- 4. STATES 6083
- 5. COMPACT CONVEX SETS 7396
- 6. STATE SPACES 94117
- 7. REPRESENTATION BY AFFINE CONTINUOUS FUNCTIONS 113136
- 8. GENERAL COMPARABILITY 126149
- 9. DEDEKIND σ-COMPLETENESS 141164
- 10. CHOQUET SIMPLICES 153176
- 11. AFFINE CONTINUOUS FUNCTIONS ON CHOQUET SIMPLICES 166189
- 12. METRIC COMPLETIONS 188211
- 13. AFFINE CONTINUOUS FUNCTIONS ON STATE SPACES 207230
- 14. SIMPLE DIMENSION GROUPS 217240
- 15. NORM-COMPLETENESS 236259
- 16. COUNTABLE INTERPOLATION AND MONOTONE σ- COMPLETENESS 263286
- 17. EXTENSIONS OF DIMENSION GROUPS 285308
- EPILOGUE: FURTHER K-THEORETIC APPLICATIONS 309332
- OPEN PROBLEMS 317340
- BIBLIOGRAPHY 325348
- INDEX 333356