Partially ordered abelian groups with interpolation / K.R. Goodearl.

Format:

Book

Author/Creator:

Goodearl, K. R., author.

Series:

Mathematical surveys and monographs ; no. 20.

Mathematical surveys and monographs, 0076-5376 ; number 20

Language:

English

Subjects (All):

Abelian groups.

Interpolation.

Physical Description:

1 online resource (360 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1986]

Summary:

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 $C DEGREES*$-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. This is a reprint of the 1986 original. (SUR

Contents:

""Contents""; ""Preface""; ""Prologue: Partially Ordered Grothendieck Groups""; ""Notational Conventions""; ""1. Basic Notions""; ""Partially ordered abelian groups""; ""Infima and suprema""; ""Ideals and quotient groups""; ""Categories of partially ordered abelian groups""; ""Pullbacks, pushouts, and coproducts""; ""Additional concepts""; ""2. Interpolation""; ""Riesz interpolation and decomposition properties""; ""Ideals and quotient groups""; ""Extensions""; ""Products, pullbacks, and pushouts""; ""2-unperforated interpolation groups""; ""Relatively bounded homomorphisms""

""3. Dimension Groups""""Dimension groups""; ""Products, pullbacks, and pushouts""; ""Simplicial groups""; ""Direct limits of simplicial groups""; ""4. States""; ""Existence""; ""Values of states""; ""Uniqueness""; ""Additional uniqueness criteria""; ""Discrete states""; ""5. Compact Convex Sets""; ""Basic definitions""; ""Categorical concepts""; ""Extreme points and faces""; ""Separation by hyperplanes""; ""Existence of extreme points""; ""Probability measures""; ""Faces of probability measures""; ""6. State Spaces""; ""Basic structure""; ""Some examples""; ""Functoriality""

""Products and limits""""Faces""; ""Change of order-unit""; ""Discrete states""; ""7. Representation By Affine Continuous Functions""; ""Affine continuous function spaces""; ""Affine representations""; ""Order-unit norms""; ""Bounded homomorphisms""; ""8. General Comparability""; ""Characteristic elements""; ""Projection bases""; ""Comparability""; ""Extremal states""; ""Closures of faces""; ""Functional representations""; ""9. Dedekind Completeness""; ""Prototypical examples""; ""Additional examples""; ""General comparability""; ""Functional representations""; ""10. Choquet Simplices""

""Simplices""""Faces""; ""Complementary faces""; ""Choquet simplices""; ""Categorical properties""; ""11. Affine Continuous Functions On Choquet Simplices""; ""Interpolation""; ""Inverse limits""; ""Semicontinuous functions""; ""Compact sets of extreme points""; ""Closed faces""; ""Complementary faces""; ""12. Metric Completions""; ""Completions with respect to positive homomorphisms""; ""Dedekind completeness""; ""Completions with respect to extremal states""; ""Criterion for extremal states""; ""Closed faces""; ""13. Affine Continuous Functions On State Spaces""; "" Approximations""

""Compact sets of extremal states""""Closed faces""; ""14. Simple Dimension Groups""; ""Simplicity""; ""State spaces""; ""Classification""; ""Finite-dimensional state spaces""; ""Finite rank""; ""15. Norm-Completeness""; ""Norm-completeness""; ""Norm-completions""; ""Quotient groups""; ""Functional representations""; ""Compact sets of extremal states""; ""Closed faces""; ""Maximal ideals""; ""16. Countable Interpolation And Monotone σ- Completeness""; ""Countable interpolation""; ""Monotone σ-completeness""; ""Norm-completeness""; ""Functional representations""

""Compact sets of extremal states""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-1247-0