Topological Invariant Means and Action of Locally Compact Semitopological Semigroups

Abstract

Let a locally compact semitopological semigroup S have a separately con- tinuous left action on a locally compact Hausdorff X. We define a jointly continuous left action of the measure algebra M(S) on the bounded Borel measure space M(X) which is an analogue of the convolution of measure alge- bras M(S). We further introduce a separately continuous left action of M(S) on the dual of a M(S)-invariant subspace A of M(X)∗ in analogue with Arens product. We consider the fixed point of this action on the set of means on A (topological S-invariant mean on A) and characterize its existence in analogue with topological right stationary, ergodic properties, Dixmier condition etc. A notion of topological (S, A)-lumpy is introduced and its relation with topolog- ical S-invariant mean on A is studied. The relation of existence of topological invariant means on a subspace of X and on X itself is also studied.