Theory of Operator Algebras II pp 363-461 | Cite as

# Structure of a von Neumann Algebra of Type III

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## Abstract

We are now going to apply the results of Chapters X and XI to the structure analysis of a von Neumann algebra *M* of type III. It turns out that the crossed product *N* = *M* ⋊_{σ} *φ* **R** relative to the modular automorphism group of a faithful semi-finite normal weight *φ* is a von Neumann algebra of type II_{∞} equipped with a faithful semi-finite normal trace τ such that τ o *θ* _{ s }, = e^{−} ^{ s } τ, *s* ∈ **R**, where *θ* is the action of **R** on *N* dual to σ^{ φ }, and that,*M* ≅ *N* ⋊_{ θ } **R**, Theorem 1.1. Furthermore, this decomposition of *M* is unique in the strongest sense. This chapter is devoted to the analysis of consequences of Theorem 1.1. In §1 we discuss general facts on this decomposition. Besides Theorem 1.1, the stability of a trace scaling one parameter automorphism group of a von Neumann algebra *N* of II_{∞}, Lemma 1 2, together with the relative commutant theorem, Theorem 1.7, will have strong implications on the structure of the automorphism group Aut(*M*) of a factor *M* of type III. The uniqueness of {*N*, **R**, *θ*) gives an immediate algebraic invariant to *M*, i.e. the flow {*C* _{ N }, **R**, *θ*} on the center *C* _{ N } of *N* is an invariant of *M*. If it is periodic with period *T* > 0, then the factor *M* is said to be of type III_{λ} with λ = e^{−2π/T }. In this case, *M* admits a discrete decomposition: *M* ≅ *N* _{0} ⋊*θ* _{0} **Z** as {*N*, **R**, *θ*} is induced from a covariant system {*N* _{0}, **Z**, *θ* _{0}}, and *N* _{0} is a factor. Section 2 is devoted to the study of such factors. If *N* is a factor, then *M* is called of type III_{1}. In this case, we don’t have a discrete decomposition of *M* in general. If the ergodic flow {*C* _{ N }, **R**, *θ*) is properly ergodic, then the factor *M* is called of type III_{0}. We will handle such factors in §3. Thanks to the Ambrose-Kakutani-Kubo-Krengel Theorem, Theorem 3.2, concerning the representation of an ergodic flow, we still have a discrete decomposition of a factor of type III_{0}, Theorem 3.7.

## Keywords

Covariant System Partial Isometry Maximal Abelian Subalgebra Faithful Normal State Infinite Multiplicity## Preview

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## Notes on Chapter XII

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