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

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 $\Bbb T$-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.