Some contributions to the theory of rings of operators. II

Abstract

In [3], the first paper in this series, we extended various results obtained by von Neumann in [6; 7] to general substantial rings.Now, in this paper, we are able to extend them still further-to arbitrary rings of operators.The plan of this paper is similar to that of [3].First, we extend the notion of coupling operator to general rings; then, using the crucial Radon-Nikodym Theorem of Dye proved in [2], we show that the coupling operator is an important invariant governing the spacial isomorphism of rings.Finally, these results are applied to questions of topology in rings; the main result in this direction being the fact that the strongest and the <r-weak topologies are purely algebraic for arbitrary rings.Another result is the fact that the notion of subring is purely algebraic.The notation used in this paper is essentially the same as that of [3], which in turn is essentially that of Murray and von Neumann in [5;6;7].However, we wish to emphasize the fact that here the symbol < between projections E and F(E < F) in a ring will denote the fact that E is equivalent to a subprojection of Fand will denote a proper ordering only when specifically stated.In preparing this paper we have received much encouragement and assistance from Professors I. E. Segal, I. Kaplansky, and R. Kadison, which we gratefully acknowledge.Note: Misonou, in [4], has proved a special case of Theorem 2 of this paper.1.The coupling operator for rings of type III.We begin by introducing some definitions and a few general lemmas.The first lemma is a useful form of Zorn's Lemma.Definition 1.1.Let @ be the set of projections possessing a property of projections in a commutative ring M. We shall call the property hereditary if the following conditions are satisfied: 1.For every collection of orthogonal projections (P|aG2l)C, we have X)"ea Pa=Pd also.2. If O^PG and Q is a projection satisfying O^Q^P, then Qd&.Lemma 1.1.A set corresponding to some hereditary property contains a maximal projection.Further, if each nonzero projection in the ring contains a nonzero member of @, then the identity operator I lies in .