Characterization of a continuous geometry within the unit group

Abstract

For a linear manifold whose endomorphism ring is not of characteristic 2, the projective geometry of subspaces is determined to within isomorphism or anti-isomorphism by the unit group of the endomorphism ring. The main purpose of this paper is to show that an analogous result applies in the case of von Neumann's continuous geometries and the associated continuous rings [1-7], and to give a characterization of the isomorphisms of the unit group of a continuous ring analogous to that known for the projective case. Our treatment of the continuous case follows rather closely the treatment of the projective case given by Baer [8], but the absence of points (i.e., of minimal elements in the lattice of subspaces) makes it necessary in the continuous case to phrase the arguments entirely in the language of ring theory and lattice theory. The main results are contained in the Structure Theorem (Theorem 7) and the Isomorphism Theorem (Theorem 6) which we prove in Part IV. After presenting some preliminary lemmas (Part I), we discuss in Part II certain properties of two kinds of elements belonging to the unit group: involutions (elements u such that u2= 1) and elements of class 2 (elements t such that (t -1) O0, (t 1)2 = 0), and show that the elements of class 2 can be characterized within the unit group (Theorem 1). In Part III, we introduce certain sets of involutions (A-systems) and characterize these within the unit group (Theorem 2). We then exhibit a 1-1 correspondence between the class of all A-systems and the class of all nonzero subspaces (Theorem 5), and show that the relation of betweenness in the lattice can be completely described in terms of A-8ystems and certain derived systems A2 which consist of elements of class 2 (Theorem 3). This provides the necessary techniques for Part IV.