Geometry of State Spaces of Operator Algebras

by Alfsen, Erik M.; Shultz, Frederic W.

• ISBN13: 9780817643195
• ISBN10: 0817643192
• Format: Hardcover
• Copyright: 2002-12-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	v
PART I: Jordan Algebras and Their State Spaces	1	(208)
JB-algebras	3	(34)
Introduction	3	(13)
Quotients of JB-algebras	16	(6)
Projections and compressions in JB-algebras	22	(4)
Operator commutativity	26	(4)
Order derivations on JB-algebras	30	(2)
Appendix: Proof of Proposition 1.3	32	(2)
Notes	34	(3)
JBW-algebras	37	(42)
Introduction	37	(5)
Orthogonality and range projections	42	(4)
Spectral resolutions	46	(2)
The lattices of projections and compressions	48	(6)
σ-weakly closed hereditary subalgebras	54	(1)
Center of a JBW-algebra	55	(3)
The bidual of a JB-algebra	58	(4)
The predual of a JBW-algebra	62	(8)
JW-algebras	70	(3)
Order automorphisms of a JB-algebra	73	(2)
Abstract characterization of compressions	75	(1)
Notes	76	(3)
Structure of JBW-algebras	79	(24)
Equivalence of projections	79	(4)
Comparison theory	83	(2)
Global structure of type I JBW-algebras	85	(2)
Classification of type I JBW-factors	87	(9)
Atomic JBW-algebras	96	(5)
Notes	101	(2)
Representations of JB-algebras	103	(36)
Representations of JBW-factors as JW-algebras	103	(5)
Representations of JB-algebras as Jordan operator algebras	108	(5)
Reversibility	113	(7)
The universal C*-algebra of a JB-algebra	120	(7)
Cartesian triples	127	(10)
Notes	137	(2)
State Spaces of Jordan Algebras	139	(52)
Carrier projections	139	(4)
The lattice of projective faces	143	(1)
Dual maps of Jordan homomorphisms	144	(3)
Traces on JB-algebras	147	(10)
Norm closed faces of JBW normal state spaces	157	(11)
The Hilbert ball property	168	(4)
The σ-convex hull of extreme points	172	(6)
Trace class elements of atomic JBW-algebras	178	(5)
Symmetry and ellipticity	183	(5)
Notes	188	(3)
Dynamical Correspondences	191	(18)
Connes orientations	191	(5)
Dynamical correspondences in JB-algebras	196	(9)
Comparing notions of orientation	205	(1)
Notes	206	(3)
PART II: convexity and Spectral Theory	209	(104)
General Compressions	211	(40)
Projections in cones	211	(11)
Definition of general compressions	222	(5)
Projective units and projective faces	227	(14)
Geometry of projective faces	241	(8)
Notes	249	(2)
Spectral Theory	251	(62)
The lattice of compressions	251	(9)
The lattice of compressions when A = V*	260	(11)
Spaces in spectral duality	271	(20)
Spectral convex sets	291	(12)
Appendix. Proof of Theorem 8.87	303	(8)
Notes	311	(2)
PART III: State Space Characterizations	313	(106)
Characterization of Jordan Algebra State Spaces	315	(42)
The pure state properties	315	(4)
The Hilbert ball property	319	(4)
Atomicity	323	(2)
The type I factor case	325	(14)
Characterization of state spaces of JB-algebras	339	(4)
Characterization of normal state spaces of JBW algebras	343	(11)
Notes	354	(3)
Characterization of Normal State Spaces of von Neumann Algebras	357	(18)
The normal state space of B(H)	357	(1)
3-frames, Cartesian triples, and blown up 3-balls	358	(7)
The generalized 3-ball property	365	(8)
Notes	373	(2)
Characterization of C*-algebra State Spaces	375	(44)
States and representations of JB-algebras	376	(6)
Reversibility of JB-algebras of complex type	382	(11)
The state space of the universal C*-algebra of a JB-algebra	393	(9)
Orientability	402	(4)
Characterization of C*-algebra state spaces among JB-algebra state spaces	406	(8)
Characterization of C*-algebra state spaces among convex sets	414	(3)
Notes	417	(2)
Appendix	419	(32)
Bibliography	451	(8)
Index	459