Averaging sequences and modulated ergodic theorems for weakly almost periodic group representations

Let T be a weakly almost periodic (WAP) representation of a locally compact σ-compact group G by linear operators in a Banach space X, and let M = M(T) be its ergodic projection onto the space of fixed points (i.e., Mχ is the unique fixed point in the closed convex hull of the orbit of χ). A sequence of probabilities {μ_n} is said to average T [weakly] if ∫ T(t)χ dμ_n converges [weakly] to M(T)χ for each χ ∈ X. We call {μ_n} [weakly] unitarily averaging if it averages [weakly] every unitary representation in a Hilbert space, and [weakly] WAPR-averaging if it averages [weakly] every WAP representation. We investigate some of the relationships of these notions, and connect them with properties of the regular representation (by translations) in the space WAP(G).