Decompositions of von Neumann lattices

Abstract

We shall use the term von Neumann lattice to refer to a complete complemented modular lattice with continuous operations [Axioms I-V, I], and reserve the term continuous geometry for irreducible von Neumann lattices [Axiom VI, 11. A satisfactory decomposition theory for von Neumann lattices would give a structure theory in terms of continuous geometries and complete Boolean algebras, their centers. Von Neumann considered this problem [pp. 259263, l] and showed that a von Neumann lattice L is isomorphic to a direct product of continuous geometries iff the center of L is atomic. In [2] and [3], Israel Halperin constructed what might be called continuous direct sums of constant lattices over complete Boolean algebras. It would be nice if every von Neumann lattice were a direct product of such lattices, but this is not the case, even for measure algebra centers, as is shown by an example in Section 7. That example is constructed by using a continuously varying family of algebraic extensions of the rational numbers. The methods used in this paper are suggested by direct integral theory and are closely related to the methods used by T. Iwamura in [4] and [5] and by I. Halperin in [2] and [3]. Many of the results are additions to the results of Iwamura and Halperin. The first section begins with a method of obtaining the subdirect decompositions of Iwamura [4], [5]. The second section contains examples to show that the most obvious attempt to make a constructive theory which reverses the Iwamura decomposition is not suitable. The third section then is devoted to a more suitable construction. Section 4 deals with the behavior of centers in continuous direct sums and Sections 5, 6, and 7 are devoted to more detailed information about more specific cases: measure algebras and continuous sums of projectiv-e geometries.