An unexpected asymmetry in the ℓ²-cohomology of amenable groups

A function f on a countable group G forms a cocycle for the left regular representation of G if for every g in G, the difference between f and its left g-translate belongs to ℓ²(G). I will show that although the left and right regular representations are unitarily equivalent in general, their corresponding spaces of cocycles can be genuinely different. Using special asymmetric cocycles, I will explain how to construct Bernoulli schemes over amenable groups whose left dynamics differ substantially from their right dynamics. This is joint work with Zemer Kosloff.