Some contributions to the theory of rings of operators

Abstract

which some of my results are announced.Chapter I.The coupling operator 1.1.Introduction.In [5, Section 10], there is introduced a finite real constant C which relates a factor M and its commutant M'.This constant later turns out to have important properties with respect to unitary equivalence.It is the purpose of this chapter to extend the notion of this invariant C to general rings of operators.In later chapters, it will be shown that quite a number of properties of rings depend on this invariant.As this invariant turns out to be an operator in the center of M, it is termed the coupling operator.(For a similar operator, defined only for finite rings, see [3].)Before getting down to the task of defining C we first prove some lemmas of general usefulness.Lemma 1.1.1.Let M be a ring with commutant M'.If E, E' are projections in M, M' respectively, such that EE' = 0, then there exists a projection P in the center of M such that Eg>P, E'gP1.Proof.Consider M= [x|__'il_"'x = 0].For xGM; AGM, A'GM': E'M'Ax = AE'M'x = 0 and E'M'A'xQE'M'x = 0, showing that AM and A'MQM.Thus nMC\M' or Pjft = P is a projection in the center of M. HxGH; E'M'Ex = EE'M'x = 0, from EE'= 0. Thus EHQM.Now consider EP and E'P: If x is arbitrary in H, EPx = Ex as ExGSW = PH.Thus EP = E. Also 'P = 0 as E"M = Q.Definition 1.1.A ring M is said to be countably decomposable if every collection {Ea} (aG-0 of orthogonal projections in M is countable.Lemma 1.1.2.Let w be a positive linear functional on a ring M such that if <_(P) = 0 for a projection E in M, then E = 0. M is now countably decomposable.Proof.Let Ea, <~Gr, be any orthogonal family of projections in M. Let 4 be the set of all a'sGT such that 1/ik + l) <co(_a) 1/k where k is a positive integer.(Assume that w(J) = 1.)Clearly V = Ulifc.However, k contains at most k a's since if it could contain k + 1 a's, say ai, , ak+i, then X- G)iEai)> ik + l)/ik + l) = 1, which is impossible as X- uiE) -( _D*_i E() CT) = 1-Thus, T is a countable union of sets of finite cardinality, and therefore is countable itself.Definition 1.2.A positive linear functional w on a ring M is termed countably additive if for any countable collection En o orthogonal projections