Geometry of State Spaces of Operator Algebras (Mathematics: Theory & Applications)

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 0001......Page 2 0002......Page 3 0003......Page 4 0004......Page 5 0005......Page 6 0006......Page 7 0007......Page 8 0008......Page 9 0009......Page 10 0010......Page 11 0011......Page 12 0012......Page 13 0013......Page 14 0014......Page 15 0015......Page 16 0016......Page 17 0017......Page 18 0018......Page 19 0019......Page 20 0020......Page 21 0021......Page 22 0022......Page 23 0023......Page 24 0024......Page 25 0025......Page 26 0026......Page 27 0027......Page 28 0028......Page 29 0029......Page 30 0030......Page 31 0031......Page 32 0032......Page 33 0033......Page 34 0034......Page 35 0035......Page 36 0036......Page 37 0037......Page 38 0038......Page 39 0039......Page 40 0040......Page 41 0041......Page 42 0042......Page 43 0043......Page 44 0044......Page 45 0045......Page 46 0046......Page 47 0047......Page 48 0048......Page 49 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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.

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 topic of this book is the theory of state spaces of operator algebras and their geometry. The states are of interest because they determine representations of the algebra, and its algebraic structure is in an intriguing and fascinating fashion encoded in the geometry of the state space. From the beginning the theory of operator algebras was motivated by applications to physics, but recently it has found unexpected new applica­ tions to various fields of pure mathematics, like foliations and knot theory, and (in the Jordan algebra case) also to Banach manifolds and infinite di­ mensional holomorphy. This makes it a relevant field of study for readers with diverse backgrounds and interests. Therefore this book is not intended solely for specialists in operator algebras, but also for graduate students and mathematicians in other fields who want to learn the subject. We assume that the reader starts out with only the basic knowledge taught in standard graduate courses in real and complex variables, measure theory and functional analysis. We have given complete proofs of basic results on operator algebras, so that no previous knowledge in this field is needed. For discussion of some topics, more advanced prerequisites are needed. Here we have included all necessary definitions and statements of results, but in some cases proofs are referred to standard texts. In those cases we have tried to give references to material that can be read and understood easily in the context of our book. 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 Book Provides 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). This Material, Which Has Previously Only Appeared In Research Papers Based On Substantial Prerequisites, Is Made Accessible To A Broad Mathematical Audience.

the Necessary Prerequisites On C*-algebras And Von Neumann Algebras Are Summarized In An Appendix, Which Contains All Relevant Definitions And Results. All Necessary Background On Jordan Algebras Is Given In The Present Book, And The First Part Of The Book May Be Of Independent Interest For Those Wanting A Quick Introduction To Jordan Algebras. This Work Will Be Of Interest To Researchers And Students In Operator Algebras.

"In the authors' previous book, State Spaces of Operator Algebras: Basic Theory, Orientations and C[superscript *]-products, the role of orientations was examined and all the prerequisites on C[superscript *]-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 mathematicians in other areas."--Jacket This chapter starts with the definition and basic properties of JB-algebras.