Von Neumann and Lattice Theory

1. Introduction. John von Neumann's brilliant mind blazed over lattice theory like a meteor, during a brief period centering around 1935-1937. With the aim of interesting him in lattices, I had called his attention, in 1933-1934, to the fact that the sublattice generated by three subspaces of Hubert space (or any other vector space) contained 28 subspaces in general, to the analogy between dimension and measure, and to the characterization of projective geometries as ir-reducible, finite-dimensional, complemented modular lattices. As soon as the relevance of lattices to linear manifolds in Hubert space was pointed out, he began to consider how he could use lattices to classify the factors of operator-algebras. One can get some impression of the initial impact of lattice concepts on his thinking about this classification problem by reading the introduction of [62J, 1 in which a systematic lattice-theoretic classification of the different possibilities was initiated. In particular, he saw that factors of Type Hi gave rise to continuous-dimensional modular subspace lattices. However, von Neumann was not content with considering lattice theory from the point of view of such applications alone. With his keen sense for axiomatics, he quickly also made a series of fundamental contributions to pure lattice theory. The primary aim of the following paragraphs is to sketch these contributions.