On invariant means and applications to ergodic theory and harmonic analysis

This thesis is concerned with the existence and properties of invariant means on certain Banach spaces and their applications to ergodic theory and harmonic analysis. The principal results obtained are as follows. Let G denote either a a-compact, unimodular amenable group or a countable, cancellative semigroup realized homomorphically by measure preserving transformations on a measure space (x, S, μ ) via the maps x → xg. Then there exists an increasing sequence {Sn} in G such that for all f ∊ Lp (X), 1 ⩽ p < ∞ the limit [equation here] exists in the mean of order p and almost everywhere. If G is an amenable topological semigroup then it has been shown by H.A. Dye that the ergodic mixing theorem is valid for G. It is proved that the amenability condition can be entirely removed and a mixing theorem is obtained, valid for arbitrary topological semigroups. The idea of an invariant mean can be dualized to invariant means on the von Neumann algebra of a group. The existence and in general non-uniqueness of such means is proved. The group von Neumann algebra is also used to show that a classical theorem of Bochner may be rephrased so as to become valid for arbitrary amenable groups rather than Abelian groups