Averaging sequences and modulated ergodic theorems for weakly almost periodic group representations (Q1293565)

Let \(T\) be a weakly almost periodic (WAP) representation of a locally compact \(\sigma\)-compact group \(G\) by linear operators in a Banach space \(X\). Let \(Mx\) be the unique fixed point in the closed convex hull of the orbit of \(x\) under \(T\). A sequence \(\{\mu_n\}\) of probabilities is said to average \(T\) [weakly] if \(\int T(t)x\mu_n(dt)\) converges [weakly] to \(Mx\) for each \(x\in X\). \(\{\mu_n\}\) is called [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. The authors study relationships between these concepts and explore connections with properties of the regular representation by translations in the space WAP \((G)\). One of the main theorems: Let \(\{\mu_n\}\) be WAPR-averaging and \(\limsup\int| f|^rd\mu_n< \infty\) for some \(r>1\). Then \(\int f(t)T(t) x\mu_n(dt)\) converges strongly for every WAP representation \(T\) in a Banach space if and only if \(\int f(t)g(t)\mu_n(dt)\) converges for every \(g\in AP(G)\). -- There is a similar theorem for unitary representations. The authors also study pointwise averaging sequences.