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Softcover ISBN:  9780821849804 
Product Code:  SURV/20.S 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470412470 
Product Code:  SURV/20.S.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9780821849804 
eBook ISBN:  9781470412470 
Product Code:  SURV/20.S.B 
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MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 20; 1986; 336 ppMSC: Primary 06; Secondary 16; 18; 19; 46;
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 crosscultural 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 firstyear 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 doubledual representation of an interpolation group with orderunit via affine continuous realvalued functions on its state space; the structure of dimension groups complete with respect to the orderunit norm, as well as monotone sigmacomplete 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

Chapters

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. Normcompleteness

16. Countable interpolation and monotone $\sigma $completeness

17. Extensions of dimension groups


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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 crosscultural 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 firstyear 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 doubledual representation of an interpolation group with orderunit via affine continuous realvalued functions on its state space; the structure of dimension groups complete with respect to the orderunit norm, as well as monotone sigmacomplete 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.

Chapters

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. Normcompleteness

16. Countable interpolation and monotone $\sigma $completeness

17. Extensions of dimension groups