Classification of outer actions of discrete amenable groupoids on injective factors

TL;DR

This paper classifies outer actions of discrete amenable groupoids on injective factors using a unified approach and cohomology reduction, avoiding traditional resolution group methods.

Contribution

It introduces a novel classification method for outer actions of discrete amenable groupoids on injective factors, bypassing the need for resolution group techniques.

Findings

Successful classification of outer actions on injective factors.

Development of a cohomology reduction theorem for discrete amenable equivalence relations.

Elimination of reliance on Katayama-Takesaki resolution groups.

Abstract

We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama-Takesaki type resolution group approach.