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.

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.