Means on semigroups and the Hahn-Banach extension property

Abstract

on the range space, the class of semigroups which permit the two types of invariant extensions is contained in the class of functions which have invariant means definable on the associated Banach space of bounded, real-valued functions defined on these semigroups. Further it was shown that every semigroup known to have an invariant mean also permitted the two types of extensions. When structure in addition to the minimal required structure is placed on the range space, e.g., when the space is an ordered subspace of the conjugate space of an ordered linear space with reproducing cone, a semi-group which has an invariant mean also has the two extension properties relative to these particular range spaces. Many of the standard function spaces satisfy these conditions. In addition this paper will exhibit conditions which guarantee the continuity of these extensions in the case when linear topological spaces are considered and make some application to real-valued functions. ?11 outlines definitions and results preceding this paper. ?III contains theorems on the existence of invariant extensions for a given semigroup G with an invariant mean and a given range space V. Various extra conditions added to the known necessary condition that V be a boundedly complete vector lattice are sufficient for these extensions. Some converse results are also given under which the existence of some sort of invariant extension implies that G has an invariant mean. ?IV contains examples of spaces satisfying the conditions imposed on the space V. ?V discusses continuity of the extensions when order and topology are both present. ?VI presents applications of the theorems of ??III and V to some of the examples of ?IV.