Abstract
The first section of this chapter presents the elements of the structure theory of von Neumann algebras with precise definitions and theorems, but with proofs referred to standard texts. The second section contains basic results on symmetries in a von Neumann algebra and the associated reflections of its normal state space, together with the necessary technical results on pairs of non-commuting projections (sometimes referred to as “non-commutative trigonometry”). The third section starts out with a general discussion of the order derivations of a von Neumann algebra (defined in Chapter 1) which leads up to a representation theorem for the rotational derivations, relating these particular order derivations to symmetries (Theorem 6.76), followed by a description of the associated “rotational motion” of the normal state space.
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