State Spaces of Operator Algebras

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

• ISBN13: 9780817638900
• ISBN10: 0817638903
• Format: Hardcover
• Copyright: 2001-05-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Preface	v
Introduction	1	(62)
Basic notions from convexity and ordered vector spaces	1	(8)
Order unit and base norm spaces	9	(8)
Selected topics in functional analysis	17	(9)
Spectral theory for monotone complete CR(X)	26	(4)
Elementary dimension theory in lattices	30	(3)
Ordered algebras	33	(9)
Algebras with involution	42	(13)
Order derivations	55	(7)
Notes	62	(1)
Elementary Theory of C*-algebras and von Neumann Algebras	63	(67)
Basics on C*-algebras	63	(15)
Representations of C*-algebras	78	(7)
Preliminaries on B(H)	85	(16)
Basics on von Neumann algebras	101	(20)
Miscellaneous	121	(8)
Notes	129	(1)
Ideals, Faces and Compressions	130	(46)
Projections, ideals, and faces for von Neumann algebras	130	(21)
Projections, ideals and faces for C*-algebras	151	(10)
Invariant subspaces	161	(7)
Compressions of von Neumann algebras	168	(6)
Notes	174	(2)
The Normal State Space of B(H)	176	(23)
Facial Structure	176	(7)
The concepts of angle and geodesic metric	183	(6)
*-isomorphisms and *-anti-isomorphisms	189	(4)
Orientation of balls and multiplication in M2(C)	193	(5)
Notes	198	(1)
States, Representations, and Orientations of C*-algebras	199	(42)
State space geometry and representations	200	(8)
The spectrum and primitive ideal space	208	(5)
Completely positive maps	213	(2)
Orientations of state spaces	215	(8)
Orientations and C* structures	223	(16)
Orientations and non-unital C*-algebras	239	(1)
Notes	240	(1)
Symmetries and Rotations in von Neumann Algebras	241	(40)
Elements of structure theory	241	(8)
Symmetries and reflections	249	(16)
Rotational Derivations	265	(14)
Notes	279	(2)
Orientations and von Neumann Algebras	281	(60)
Balanced symmetries and associative products	282	(13)
Cartesian triples and 3-frames	295	(19)
Orientation of normal state spaces	314	(15)
From orientations to associative products	329	(11)
Notes	340	(1)
Bibliography	341	(4)
Index	345