The flow of weights on factors of type ${\rm III}$

Abstract

Chapter 0. Introduction and Preliminary 473 Chapter I.The Global Flow of Weights 476 1.0.Introduction 476 1.1.Elementary Comparison of Weights 477 1.2.The Global Flow of Weights on Factors of Type III;, λφl... .486Chapter II.Integrable Weights on Factors of Type III 494 Π.O.Introduction 494 11.1.Dominant Weights 495 Π.2.Integrable Weights and the Smooth Flow of Weights 500 Π.3.Computation of the Smooth Flow of Weights (1) 505 Π.4.Regularization of Weights of Infinite Multiplicity 508 Π.5.Relative Commutant Theorem 513 Π.6.Computation of the Smooth Flow of Weights (2) 517 Chapter III.Non-Abelian Cohomology in Properly Infinite von Neumann Algebras 524 111.0.Introduction 524 111.1.Elementary Properties of Twisted ^-Representation 525 111.2.Tensor Product and Integrability of Twisted ^Representations 527 111.3.Integrable Actions Duality and Invariant Γ 535 111.4.Galois Correspondence 540 111.5.Stability of Automorphisms 544 Chapter IV.The Flow of Weights and the Automorphism Group of a Factor of type III 548 IV.O.Introduction 548 IV.l.The Fundamental Homomorphism 549 IV.2.The Extended Modular Automorphism Groups 556 IV.3.The Exact Sequence for the Group of all Automorphisms 558Introduction.The recent developments in the theory of operator algebras showed the importance of the study of weights.A weight φ on a von Neumann algebra M is a linear map from M+ to [0, +<*>]; φ is faithful if φ(x) -0 implies x -0, normal if it commutes with the sup operation, semi-finite when φ(x)< + co on a cr-strongly dense subset of M + .Throughout this paper, a weight means a semi-finite normal one.