Geometry of State Spaces of Operator Algebras / Edition 1

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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.

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Editorial Reviews

From the Publisher

From the reviews:

"The two books together provide a predominantly self-contained presentation of the geometric theory of operator algebra state spaces, culminating in the classification theorem of Alfsen, Hanche–Olsen and Shultz. Until now much of this material has been accessible only in the original papers, which makes the two volumes a welcome addition to the literature. . . . The result is a clear and comprehensive account. . . . the book describes a beautiful solution to a problem dating back to the foundations of the subject."


"Notable results…are presented in this book in a unified way, with complete and enlightening proofs and comments. The authors have done fine work for the mathematical community, providing a valuable toolkit for researchers interested in non-associative structures, self-adjoint operator algebras, or areas of functional analysis or mathematical physics where aspects related to convexity and ordered spaces appear…."


"The aim of the present book is to give a complete geometric description of the state spaces of operator algebras, meaning to give axiomatic characterizations of those convex sets that are state spaces … . The book is divided into three parts. … It is aimed to specialists in operator algebras, graduate students and mathematicians working in other areas (mathematical physics, foundation of quantum mechanics).” (S. Cobzas, Mathematica, Vol. 46 (2), 2004)

"The authors of this monograph present a complete and self-contained solution to the long-standing problem of giving a geometric description of state spaces … . There also are an Appendix, a Bibliography containing 137 references, and an Index. The material, which previously has appeared only in research papers … is made accessible here to a broad mathematical audience. … The book under review is intended for specialists in operator algebras, as well as graduate students and mathematicians in other areas.” (Radu Iordanescu, Revue Roumaine de Mathématiques Pures et Appliquées, Vol. XLIX (3), 2004)

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Product Details

  • ISBN-13: 9780817643195
  • Publisher: Birkhauser Verlag
  • Publication date: 12/13/2002
  • Series: Mathematics: Theory & Applications Series
  • Edition description: 2003
  • Edition number: 1
  • Pages: 467
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.06 (d)

Table of Contents

Pt. I Jordan Algebras and Their State Spaces 1
Ch. 1 JB-algebras 3
Ch. 2 JBW-algebras 37
Ch. 3 Structure of JBW-algebras 79
Ch. 4 Representations of JB-algebras 103
Ch. 5 State Spaces of Jordan Algebras 139
Ch. 6 Dynamical Correspondences 191
Pt. II Convexity and Spectral Theory 209
Ch. 7 General Compressions 211
Ch. 8 Spectral Theory 251
Pt. III State Space Characterizations 313
Ch. 9 Characterization of Jordan Algebra State Spaces 315
Ch. 10 Characterization of Normal State Spaces of von Neumann Algebras 357
Ch. 11 Characterization of C[superscript *]-algebra State Spaces 375
Appendix 419
Bibliography 451
Index 459
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