Von Neumann lattices

Abstract

In this chapter the basic properties of von Neumann lattices are described. The von Neumann lattices are the projection lattices of von Neumann algebras, and the von Neumann algebras are those *-subalgebras of B(H) which are closed with respect to certain topologies weaker than the uniform topology in B(H) Section 6.1 gives the definition of a von Neumann algebra together with the central theorem, von Neumann’s double commutant theorem (Proposition 6.1), which characterizes the closedness of *-subalgebras of B(H) (and thereby the von Neumann algebras) purely in algebraic terms. In the Section 6.2 it is proved that the set of projections of a von Neumann algebra is a complete orthomodular lattice (Proposition 6.3). This section also describes briefly the (by now) classic dimension theory of projections of a von Neumann algebra, which is intimately related to the classification theory of von Neumann algebras. Particular attention will be paid to the finite von Neumann algebras (which are not the same as the finite dimensional algebras — although finite diensional algebras are finite). The surprizing fact about a finite algebra is that its projection lattice is not only orthomodular but also modular (Proposition 6.14). The other important proposition in this section is Proposition 6.5, which gives equivalent characterization of the finiteness of an algebra in terms of existence of traces on the algebra.