Partially ordered Abelian groups with interpolation.

*(English)*Zbl 0589.06008
Mathematical Surveys and Monographs, 20. Providence, R.I.: American Mathematical Society (AMS). xxii, 336 p. (1986).

The purpose of the present book is to provide a solid foundation in the theory of partially ordered abelian groups satisfying the Riesz interpolation property (”interpolation groups”). Although interpolation groups are defined as purely algebraic structures, their development has been strongly influenced by functional analysis. The intention of the author is to make the subject accessible to readers from each culture, by providing sufficient details for both the algebraic and analytic aspects of the theory.

The first four chapters of the book are purely algebraic, beginning with an introduction to partially ordered abelian groups and in particular to interpolation groups. A partially ordered abelian group \(G\) is called interpolation group if it satisfies the following condition (a) given any \(x_1,x_2,y_1,y_2\) in \(G\) such that \(x_ i\leq y_ j\) for all i,j, there exists \(z\in G\) such that \(x_i\leq z\leq y_j\) for all \(i,j\).

In Chapter 3 the “dimension groups” are considered which may be described as upward directed interpolation groups in which positive integers can be cancelled from inequalities. There is a structural characterization of dimension groups as direct limits of finite products of copies of the integers. Chapter 4 is concerned with states on a partially ordered abelian group with an order-unit. Existence theorems for states, formulas for ranges of values of states on group elements, and conditions for uniqueness of states are developed.

In Chapter 5, a purely functional analytic interlude occurs, designed to introduce the more algebraic reader to some basic aspects of compact convex sets including the Krein-Mil’man Theorem. The following two chapters introduce the connection through which functional analysis is brought to bear on interpolation groups. Chapter 6 is concerned with the state space on a partially ordered group with an order-unit and with the structure of the state space as a compact convex subset of the linear topological space of all real-valued functions on the group, while Chapter 7 is concerned with the natural representation of the group as affine continuous real-valued functions on the state space.

A long-range goal is to describe any interpolation group with order-unit as completely as possible in terms of affine continuous functions on its state space. The cases in which this goal is most readily accessible are developed in Chapters 8 and 9, namely the cases of interpolation groups with sufficiently large supply of direct sum decompositions, and Dedekind \(\sigma\)-complete lattice ordered abelian groups. There is also a complete description of any Dedekind \(\sigma\)-complete lattice ordered abelian group with order-unit in terms of continuous real-valued functions on a basically disconnected Hausdorff space.

In Chapters 10 and 11 the author gives the remaining portion of functional analysis needed in the book, namely some of the general properties of Choquet simplices and their affine continuous function spaces are developed. Chapter 12 concerns completions of interpolation groups with respect to pseudo-metrics derived from states.

In the remainder of the book, some of the fruits of the combined algebraic and analytic developments appear. For example, a characterization of the closure of the image of the natural affine continuous function representation of an interpolation group with order- unit is obtained in Chapter 13. An effective application of this result is the development of a complete description and classification of all ”simple” dimension groups in terms of Choquet simplices which is given in Chapter 14.

Chapter 15 is concerned with the class of those interpolation groups which are complete with respect to the natural norm derived from their states. In Chapter 16, which may be viewed as a corollary of the preceding chapter, the results from the norm complete case find application to groups which either satisfy the countably infinite version of the interpolation property or are monotone \(\sigma\)-complete.

The final Chapter 17 is a signpost indicating one of the directions in which open problems may be found. It is an introduction to the problem of classifying dimension groups arising as extensions of a given dimension group by another and also serves to present some techniques for constructing new dimension groups as extensions of old ones. A list of open problems concerning interpolation groups is given following the Epilogue.

The first four chapters of the book are purely algebraic, beginning with an introduction to partially ordered abelian groups and in particular to interpolation groups. A partially ordered abelian group \(G\) is called interpolation group if it satisfies the following condition (a) given any \(x_1,x_2,y_1,y_2\) in \(G\) such that \(x_ i\leq y_ j\) for all i,j, there exists \(z\in G\) such that \(x_i\leq z\leq y_j\) for all \(i,j\).

In Chapter 3 the “dimension groups” are considered which may be described as upward directed interpolation groups in which positive integers can be cancelled from inequalities. There is a structural characterization of dimension groups as direct limits of finite products of copies of the integers. Chapter 4 is concerned with states on a partially ordered abelian group with an order-unit. Existence theorems for states, formulas for ranges of values of states on group elements, and conditions for uniqueness of states are developed.

In Chapter 5, a purely functional analytic interlude occurs, designed to introduce the more algebraic reader to some basic aspects of compact convex sets including the Krein-Mil’man Theorem. The following two chapters introduce the connection through which functional analysis is brought to bear on interpolation groups. Chapter 6 is concerned with the state space on a partially ordered group with an order-unit and with the structure of the state space as a compact convex subset of the linear topological space of all real-valued functions on the group, while Chapter 7 is concerned with the natural representation of the group as affine continuous real-valued functions on the state space.

A long-range goal is to describe any interpolation group with order-unit as completely as possible in terms of affine continuous functions on its state space. The cases in which this goal is most readily accessible are developed in Chapters 8 and 9, namely the cases of interpolation groups with sufficiently large supply of direct sum decompositions, and Dedekind \(\sigma\)-complete lattice ordered abelian groups. There is also a complete description of any Dedekind \(\sigma\)-complete lattice ordered abelian group with order-unit in terms of continuous real-valued functions on a basically disconnected Hausdorff space.

In Chapters 10 and 11 the author gives the remaining portion of functional analysis needed in the book, namely some of the general properties of Choquet simplices and their affine continuous function spaces are developed. Chapter 12 concerns completions of interpolation groups with respect to pseudo-metrics derived from states.

In the remainder of the book, some of the fruits of the combined algebraic and analytic developments appear. For example, a characterization of the closure of the image of the natural affine continuous function representation of an interpolation group with order- unit is obtained in Chapter 13. An effective application of this result is the development of a complete description and classification of all ”simple” dimension groups in terms of Choquet simplices which is given in Chapter 14.

Chapter 15 is concerned with the class of those interpolation groups which are complete with respect to the natural norm derived from their states. In Chapter 16, which may be viewed as a corollary of the preceding chapter, the results from the norm complete case find application to groups which either satisfy the countably infinite version of the interpolation property or are monotone \(\sigma\)-complete.

The final Chapter 17 is a signpost indicating one of the directions in which open problems may be found. It is an introduction to the problem of classifying dimension groups arising as extensions of a given dimension group by another and also serves to present some techniques for constructing new dimension groups as extensions of old ones. A list of open problems concerning interpolation groups is given following the Epilogue.

Reviewer: Sh. A. Ayupov (Tashkent)

##### MSC:

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

46A45 | Sequence spaces (including Köthe sequence spaces) |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

16E20 | Grothendieck groups, \(K\)-theory, etc. |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

46L30 | States of selfadjoint operator algebras |

46A55 | Convex sets in topological linear spaces; Choquet theory |