On the cohomology of weakly almost periodic group representations

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.