گروه‌های ابلیان جزئی با درون‌یابی

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. CONTENTS 6 PREFACE 10 PROLOGUE: PARTIALLY ORDERED GROTHENDIECK GROUPS 14 NOTATIONAL CONVENTIONS 22 1. BASIC NOTIONS 24 Partially ordered abelian groups 24 Infima and suprema 28 Ideals and quotient groups 31 Categories of partially ordered abelian groups 34 Pullbacks, pushouts, and coproducts 38 Additional concepts 41 2. INTERPOLATION 45 Riesz interpolation and decomposition properties 45 Ideals and quotient groups 49 Extensions 51 Products, pullbacks, and pushouts 54 2-unperforated interpolation groups 58 Relatively bounded homomorphisms 60 3. DIMENSION GROUPS 67 Dimension groups 67 Products, pullbacks, and pushouts 69 Simplicial groups 70 Direct limits of simplicial groups 73 4. STATES 83 Existence 83 Values of states 86 Uniqueness 89 Additional uniqueness criteria 90 Discrete states 93 5. COMPACT CONVEX SETS 96 Basic definitions 96 Categorical concepts 98 Extreme points and faces 101 Separation by hyperplanes 103 Existence of extreme points 108 Probability measures 110 Faces of probability measures 113 6. STATE SPACES 117 Basic structure 118 Some examples 120 Functoriality 124 Products and limits 125 Faces 127 Change of order-unit 129 Discrete states 131 7. REPRESENTATION BY AFFINE CONTINUOUS FUNCTIONS 136 Affine continuous function spaces 136 Affine representations 140 Order-unit norms 143 Bounded homomorphisms 146 8. GENERAL COMPARABILITY 149 Characteristic elements 150 Projection bases 152 Comparability 154 Extremal states 155 Closures of faces 159 Functional representations 161 9. DEDEKIND σ-COMPLETENESS 164 Prototypical examples 164 Additional examples 166 General comparability 170 Functional representations 172 10. CHOQUET SIMPLICES 176 Simplices 176 Faces 181 Complementary faces 183 Choquet simplices 186 Categorical properties 187 11. AFFINE CONTINUOUS FUNCTIONS ON CHOQUET SIMPLICES 189 Interpolation 190 Inverse limits 194 Semicontinuous functions 195 Compact sets of extreme points 201 Closed faces 205 Complementary faces 209 12. METRIC COMPLETIONS 211 Completions with respect to positive homomorphisms 212 Dedekind completeness 216 Completions with respect to extremal states 219 Criterion for extremal states 222 Closed faces 227 13. AFFINE CONTINUOUS FUNCTIONS ON STATE SPACES 230 Approximations 230 Compact sets of extremal states 237 Closed faces 238 14. SIMPLE DIMENSION GROUPS 240 Simplicity 241 State spaces 243 Classification 245 Finite-dimensional state spaces 250 Finite rank 252 15. NORM-COMPLETENESS 259 Norm-completeness 260 Norm-completions 261 Quotient groups 264 Functional representations 266 Compact sets of extremal states 270 Closed faces 276 Maximal ideals 279 16. COUNTABLE INTERPOLATION AND MONOTONE σ- COMPLETENESS 286 Countable interpolation 286 Monotone σ-completeness 292 Norm-completeness 295 Functional representations 298 Compact sets of extremal states 301 Closed faces 303 Quotient groups 305 17. EXTENSIONS OF DIMENSION GROUPS 308 Extensions 309 Some examples 311 Extensions with order-units 317 Some examples 319 Existence of extensions with order-units 324 EPILOGUE: FURTHER K-THEORETIC APPLICATIONS 332 OPEN PROBLEMS 340 BIBLIOGRAPHY 348 INDEX 356 A 356 B 356 C 356 D 357 E 357 F 357 G 357 I 357 K 357 I 357 M 358 N 358 O 358 P 358 Q 358 R 358 S 359 T 359 U 359 V 359 Z 359