On the cohomology of weakly almost periodic group representations

TL;DR

This paper explores the cohomology of weakly almost periodic group representations on Banach spaces, establishing vanishing results and generalizing existing theorems using fixed point and decomposition theorems.

Contribution

It introduces a cohomological framework for weakly almost periodic representations and extends known results through new vanishing theorems and generalizations.

Findings

Proves a vanishing result for the restriction map in reduced cohomology.

Generalizes theorems on continuous group cohomology.

Utilizes fixed point and decomposition theorems to extend prior work.

Abstract

We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll-Nardzewski fixed point Theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu-Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.