Outer actions of a countable discrete amenable group on approximately finite dimensional factors I, General Theory

TL;DR

This paper introduces a cohomological invariant called the modular obstruction to classify outer actions of countable discrete amenable groups on approximately finite dimensional factors, establishing a foundational theory.

Contribution

It defines the intrinsic modular obstruction invariant for outer actions and provides a classification framework for these actions on AFD factors, especially outside type III_0.

Findings

Defined the intrinsic modular obstruction invariant.

Classified outer actions using the invariant for amenable groups.

Simplified the invariant for non-type III_0 factors.

Abstract

We associate a cohomological invariant to each outer action of a group on a factor, and classify them by the invariant in the case that the group is a countable discrete amenable group and the factor is appoximately finite dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called the intrinsic modular obstruction. The invariant for an outer action alpha is given as the pull back of the intrinsic modular obstruction, which is called the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first part of the theory and presents general theory. In the case that the factor is not of type III_0, the invariant is substantially simplified. These cases and examples will be discussed in forthcoming paper.