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

Yoshikazu Katayama and Masamichi Takesaki

Vol. 243 (2009), No. 1, 57–126

DOI: 10.2140/pjm.2009.243.57

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.

Keywords

outer action, outer conjugacy, cohomology of group, asymmetrization

Mathematical Subject Classification 2000

Primary: 46L55

Secondary: 46L35, 46L40, 20J06

Milestones

Received: 4 June 2009

Accepted: 11 June 2009

Published: 1 November 2009