Some contributions to the theory of rings of operators. II

TLDR: In this article, Segal et al. extended von Neumann's work on factors to general rings of operators on Hilbert spaces of arbitrary dimension and showed that the strongest topology is purely algebraic along with the notion of semifinite subring.

Abstract: The extension of von Neumann's work on factors to general rings of operators on Hilbert spaces of arbitrary dimension has been begun by Dixmier and Kaplansky in [1 ] and [3 ] (the numbers in brackets refer to the List of References at the end of the paper). It is the purpose of this paper to extend these results still further, in particular Chapter X of [5] and Chapters I to III of [8]. The general scheme of this paper is as follows: First, the constant C of von Neumann (Chapter X of [5 ]) is extended to an operator belonging to the center of a ring of semifinite type, such a ring being one with no type III part. Next, using techniques devised by Dye and von Neumann, this operator C (termed the coupling operator) is shown to be the chief invariant governing the spatial type of a ring. Finally, these results are applied to questions of topology in rings, yielding the fact that the strongest topology is purely algebraic along with the notion of semifinite subring. Besides these main results, we obtain various subsidiary results, in particular, conditions for strong and weak continuity of *-isomorphisms, continuity of the trace in various topologies, and conjugate isomorphism of a ring with its commutant. The notation of this paper is essentially that of [5] and [7], with but a few exceptions. Throughout the paper, the notation [Mx] will denote the closure in some Hilbert space H of the family of vectors I Ax } for A in a ring M and x a fixed vector. The symbol < between projections E and F (E < F) in a ring will denote the fact that E is equivalent to a subprojection of F belonging to M. This will denote a proper projection only when specifically stated. This particular notation is used because the printer does not have the symbol used in [51 for this relationship. The symbols U and n will be used in their usual sense of set theoretic union and intersection, the remaining ones being those standard in Hilbert space theory. In preparing this paper, we have received much valuable assistance from Professors I. E. Segal, I. Kaplansky, and P. R. Halmos, which we gratefully acknowledge. Note: Since this paper was written, Professor Segal has informed me that