Antidiagonal and square averages for amenable groups

We prove that, for commuting unitary representations $U,V$ of a countable amenable group G, the averages DF=1|F|∑g∈ FUg^*Vg have a common selfadjoint weak operator limit along all left and right {} sequences, with strong* convergence along two-sided sequences. This answers Bergelson's question on commuting actions. For any unitary representation $π$, we also prove that AF=|F|⁻¹∑g∈ Fπ(g²) converges strongly along right {} sequences to a selfadjoint central operator supported on the finite type~I part of $π(G)''$; the adjoints AF^* converge strongly along left {} sequences. This settles the convergence question for square averages posed by Bergelson and Rodríguez-Martín. For measure-preserving actions of finitely generated amenable or countable virtually FC groups, we obtain limₙ1|Fₙ|∑g∈ Fₙμ(A∩ Tg²A)>0 μ(A)>0. For each ε>0, the set :μ(A∩ Tg²A)>μ(A)²-ε\ meets every Bohr neighborhood of the identity in a set of positive lower density, uniformly over {} sequences. These intersections are also left and right IP^* for virtually FC groups. We construct a unitary representation $π$ of a locally finite nilpotent group and a weakly mixing vector f for which :Reπ(g²)f,f>-ε\ has upper Banach density zero for small ε>0. We also give a weakly mixing action of a two-generated solvable group for which the IP^* conclusion fails. The recurrence results yield product sets $BB$ inside difference sets, with B of positive upper density.