State Spaces of Operator Algebras

Self-contained work, focusing on the theory of state spaces of C- algebras and von Neumann algebras. Gives a new presentation of the theory of orientation of state spaces of C-algebras that does not depend on Jordan algebras. Also presented is the theory of orientation of normal state spaces of von Neumann algebras, with complete proofs for the first time. Minimal prerequisites: knowledge from standard graduate courses in real and complex variables, measure theory, and functional analysis. Intended for specialists in operator algebras, as well as graduate students and mathematicians seeking an overview of the field.

"This excellent book was born out of the authors' successful attempts to answer questions [like] 'When is a compact convex set the state space of a C-algebra?' . . . I would regard the book as essential reading for any graduate student working in C-algebras and related areas, particularly those with an interest in geometry."

Zentralblatt Math

"A useful introduction to an elegant aspect of the theory of operator algebras which has close links to mathematical physics, as well as being of interest in its own right."

Mathematical Reviews

"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 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." ---*Analele Stiintifice ale Universitatii,,al. I. Cuza din Iasi

Autorentext
Self-contained work, focusing on the theory of state spaces of C -
algebras and von Neumann algebras. Gives a new presentation of the
theory of orientation of state spaces of C -algebras that does not
depend on Jordan algebras. Also presented is the theory of orientation
of normal state spaces of von Neumann algebras, with complete proofs
for the first time. Minimal prerequisites: knowledge from standard
graduate courses in real and complex variables, measure theory, and
functional analysis. Intended for specialists in operator algebras, as
well as graduate students and mathematicians seeking an overview of
the field.

Inhalt
Preface.- 1: Introduction.- Basic notions from convexity and ordered vector spaces.- Order unit and base norm spaces.- Selected topics in functional analysis.- Spectral theory for monotone complete CR(X).- Elementary dimension theory in lattices.- Ordered algebras.- Algebras with involution.- Order derivations.- Notes.- 2: Elementary Theory of C-algebras and von Neumann Algebras.- Basics on C-algebras.- Representations of C-algebras.- Preliminaries on $$ \mathcal{B}$$ (H).- Basics on von Neumann algebras.- Miscellaneous.- Notes.- 3: Ideals, Faces and Compressions.- Projections, ideals, and faces for von Neumann algebras.- Projections, ideals and faces for C-algebras.- Invariant subspaces.- Compressions of von Neumann algebras.- Notes.- 4: The Normal State Space of 13(H).- Facial Structure.- The concepts of angle and geodesic metric.- -isomorphisms and -anti-isomorphisms.- Orientation of balls and multiplication in M2(C).- Notes.- 5: States, Representations, and Orientations of C-algebras.- State space geometry and representations.- The spectrum and primitive ideal space.- Completely positive maps.- Orientations of state spaces.- Orientations and C structures.- Orientations and non-unital C*-algebras.- Notes.- 6: Symmetries and Rotations in von Neumann Algebras.- Elements of structure theory.- Symmetries and reflections.- Rotational Derivations.- Notes.- 7: Orientations and von Neumann Algebras.- Balanced symmetries and associative products.- Cartesian triples and 3-frames.- Orientation of normal state spaces.- From orientations to associative products.- Notes.