Outer Actions of a Discrete Amenable Group on Approximately Finite Dimensional Factors III, The Type ${\text{III}}_{\lambda }$ Case, $0<\lambda <1$, Examples

Abstract

In this last article of the series on outer actions of a countable dicrete amenable group on AFD factors, we analyze outer actions of a countable discrete free abelian group on an AFD factor of type ${\text{III}}_{\lambda }$, $0<\lambda <1$, and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a $\mathbb{T}$-valued cocycle of a torsion free abelian group, which might have been known by the group cohomologists. As the asymmetrization technique gives us a very handy criteria for coboundaries, we present it here in detail in the second section.

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