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.