Outer actions of a discrete amenable group on approximately finite-dimensional factors, III: The type IIIλcase, 0<λ <1, asymmetrization and examples

Abstract

In this last article of the series on outer actions of a countable discrete amenable group on approximately finite-dimensional factors, we analyze outer actions of a countable discrete free abelian group on an approximately finite-dimensional factor of type III λ with 0 < λ < 1 and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a T-valued cocycle of a torsion-free abelian group, which might have been known by group cohomologists. As the asymmetrization technique gives us a very handy criterion for coboundaries, we present it here in detail.