Group structure and the pointwise ergodic theorem for connected amenable groups

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R). We show how to explicitly construct sequences {U_n} of compacta in G in terms of the structural features of G which have the following property: For any "reasonable" action G × L^p(X, μ) ↓ L^p(X, μ) on an L^p space, 1 < p < ∞, and any f ∈ L^p(X, μ), the averages A_nf= 1 |U_n| ∫ U_nT_g ^-1fdg (|E|= left Haar measure inG). converge in L^p norm, and pointwise μ-a.e. on X, to G-invariant functions f* in L^p(X, μ). A single sequence {U_n} in G works for all L^p actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.