Complementation in Continuous Cohomology with Coefficients in Banach Modules

TL;DR

This paper introduces weakly uniquely stationary representations to study the complementability of closed subspaces in continuous cohomology with Banach module coefficients, extending key cohomological results for various group structures.

Contribution

It presents a new framework for analyzing complementability in continuous cohomology using weakly uniquely stationary representations, advancing the understanding of cohomological properties in Banach modules.

Findings

Extended cohomological results for nilpotent groups

Refined results for products of groups

Improved understanding of lattices in cohomology

Abstract

In this article, we introduce the concept of weakly uniquely stationary representations. This framework enables us to investigate the complementability of closed subspaces within the context of continuous cohomology with coeffcients in Banach modules. As an application, we extend and refine several cohomological results from the literature, particularly in the settings of nilpotent groups, products of groups, and lattices.