This self-contained work, focusing on the theory of state spaces of C*-algebras and von Neumann algebras, explains how the oriented state space geometrically determines the algebra. The theory of orientation of C*-algebra state spaces is presented with a new approach that does not depend on Jordan algebras, and the theory of orientation of normal state spaces of von Neumann algebras is presented with complete proofs for the first time. The theory of operator algebras was initially motivated by applications to physics, but has recently found unexpected new applications to fields of pure mathematics as diverse as foliations and knot theory. Key features include: * first and only work devoted to state spaces of operator algebras--contains much material not available in existing books * prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis * complete proofs of basic results on operator algebras presented so that no previous knowledge in the field is needed * detailed introduction develops basic tools used throughout the text * numerous chapter remarks on advanced topics of independent interest with references to the literature, or discussion of applications to physics "State Spaces of Operator Algebras" is intended for specialists in operator algebras, as well as graduate students and mathematicians seeking an overview of the field. The introduction to C*-algebras and von Neumann algebras may also be of interest in it own right for those wanting a quick introduction to basic concepts in those fields.

In this book we give a complete geometric description of state spaces of operator algebras, Jordan as well as associative. That is, we give axiomatic characterizations of those convex sets that are state spaces of C*-algebras and von Neumann algebras, together with such characterizations for the normed Jordan algebras called JB-algebras and JBW-algebras. These non associative algebras generalize C*-algebras and von Neumann algebras re spectively, and the characterization of their state spaces is not only of interest in itself, but is also an important intermediate step towards the characterization of the state spaces of the associative algebras. This book gives a complete and updated presentation of the character ization theorems of [10]' [11] and [71]. Our previous book State spaces of operator algebras: basic theory, orientations and C*-products, referenced as [AS] in the sequel, gives an account of the necessary prerequisites on C*-algebras and von Neumann algebras, as well as a discussion of the key notion of orientations of state spaces. For the convenience of the reader, we have summarized these prerequisites in an appendix which contains all relevant definitions and results (listed as (AI), (A2), ... ), with reference back to [AS] for proofs, so that this book is self-contained.

This monograph presents a complete and self-contained solution to the long-standing problem of giving a geometric description of state spaces of C*-algebras and von Neumann algebras, and of their Jordan algebraic analogs (JB-algebras and JBW-algebras). The material, which previously has appeared only in research papers and required substantial prerequisites for a reader's understanding, is made accessible here to a broad mathematical audience. Key features include: The properties used to describe state spaces are primarily of a geometric nature, but many can also be interpreted in terms of physics.There are numerous remarks discussing these connections * A quick introduction to Jordan algebras is given; no previous knowledge is assumed and all necessary background on the subject is given * A discussion of dynamical correspondences, which tie together Lie and Jordan structures, and relate the observables and the generators of time evolution in physics * The connection with Connes' notions of orientation and homogeneity in cones is explained * Chapters conclude with notes placing the material in historical context * Prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis * Excellent bibliography and index In the authors' previous book, State Spaces of Operator Algebras: Basic Theory, Orientations and C*-products (ISBN 0-8176-3890-3), the role of orientations was examined and all the prerequisites on C*- algebras and von Neumann algebras, needed for this work, were provided in detail. These requisites, as well as all relevant definitions and results with reference back to State Spaces, are summarized in an appendix, further emphasizing the self-contained nature of this work.Geometry of State Spaces of Operator Algebras is intended for specialists in operator algebras, as well as graduate students and

This self-contained work, focusing on the theory of state spaces of C*-algebras and von Neumann algebras, explains how the oriented state space geometrically determines the algebra. The theory of orientation of C*-algebra state spaces is presented with a new approach that does not depend on Jordan algebras, and the theory of orientation of normal state spaces of von Neumann algebras is presented with complete proofs for the first time. The theory of operator algebras was initially motivated by applications to physics, but has recently found unexpected new applications to fields of pure mathematics as diverse as foliations and knot theory.Key features include: * first and only work devoted to state spaces of operator algebras-- contains much material not available in existing books * prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis * complete proofs of basic results on operator algebras presented so that no previous knowledge in the field is needed * detailed introduction develops basic tools used throughout the text * numerous chapter remarks on advanced topics of independent interest with references to the literature, or discussion of applications to physics State Spaces of Operator Algebras is intended for specialists in operator algebras, as well as graduate students and mathematicians seeking an overview of the field. The introduction to C*-algebras and von Neumann algebras may also be of interest in it own right for those wanting a quick introduction to basic concepts in those fields.

The importance of convexity arguments in functional analysis has long been realized, but a comprehensive theory of infinite-dimensional convex sets has hardly existed for more than a decade. In fact, the integral representation theorems of Choquet and Bishop -de Leeuw together with the uniqueness theorem of Choquet inaugurated a new epoch in infinite-dimensional convexity. Initially considered curious and tech nically difficult, these theorems attracted many mathematicians, and the proofs were gradually simplified and fitted into a general theory. The results can no longer be considered very "deep" or difficult, but they certainly remain all the more important. Today Choquet Theory provides a unified approach to integral representations in fields as diverse as potential theory, probability, function algebras, operator theory, group representations and ergodic theory. At the same time the new concepts and results have made it possible, and relevant, to ask new questions within the abstract theory itself. Such questions pertain to the interplay between compact convex sets K and their associated spaces A(K) of continuous affine functions; to the duality between faces of K and appropriate ideals of A(K); to dominated extension problems for continuous affine functions on faces; and to direct convex sum decomposition into faces, as well as to integral for mulas generalizing such decompositions. These problems are of geometric interest in their own right, but they are primarily suggested by applica tions, in particular to operator theory and function algebras."