State Spaces of Operator Algebras, 2001
Basic Theory, Orientations, and C*-products
Mathematics: Theory & Applications Series

The topic of this book is the theory of state spaces of operator algebras and their geometry. The states are of interest because they determine representations of the algebra, and its algebraic structure is in an intriguing and fascinating fashion encoded in the geometry of the state space. From the beginning the theory of operator algebras was motivated by applications to physics, but recently it has found unexpected new applica­ tions to various fields of pure mathematics, like foliations and knot theory, and (in the Jordan algebra case) also to Banach manifolds and infinite di­ mensional holomorphy. This makes it a relevant field of study for readers with diverse backgrounds and interests. Therefore this book is not intended solely for specialists in operator algebras, but also for graduate students and mathematicians in other fields who want to learn the subject. We assume that the reader starts out with only the basic knowledge taught in standard graduate courses in real and complex variables, measure theory and functional analysis. We have given complete proofs of basic results on operator algebras, so that no previous knowledge in this field is needed. For discussion of some topics, more advanced prerequisites are needed. Here we have included all necessary definitions and statements of results, but in some cases proofs are referred to standard texts. In those cases we have tried to give references to material that can be read and understood easily in the context of our book.

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.