Geometry of State Spaces of Operator Algebras

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.

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."

―MATHEMATICAL REVIEWS

"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...."

―ZENTRALBLATT MATH

"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)