Paul Milnes

Compact right topological semigroups and generalizations of almost periodicity

Preliminaries.- The structure of compact semigroups.- Subspaces of C(S) and compactifications of S.- Fixed points and left invariant means on subspaces of C(S).- Examples.

Compactifications of semitopological semigroups II

Suppose S is a semitopological semigroup. We consider various subspaces of C ( S ) and determine what topological algebraic structure can be introduced into the spaces of means on the subspaces and into the spectra of the C *-sub-algebras of C ( S ) they generate.

Haar measure for compact right topological groups

Compact right topological groups arise in topological dynamics and in other settings. Following H. Furstenbergs seminal work on distal flows, R. Ellis and I. Namioka have shown that the compact right topological groups of dynamical type always admit a probability measure invariant under the continuous left translations; however, this invariance property is insufficient to identify a unique probability measure (in contrast to the case of compact topological groups). In the present paper, we amplify on the proofs of Ellis and Namioka to show that a right invariant probability measure on the compact right topological group G exists provided G admits an appropriate system of normal subgroups, that it is uniquely determined and that it is also invariant under the continuous left translations. Using Namiokas work, we show that G has such a system of subgroups if its topological centre contains a countable dense subset, or if it is a closed subgroup of such a group.

LOCALLY COMPACT GROUPS, INVARIANT MEANS AND THE CENTRES OF COMPACTIFICATIONS

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L ∞ ( G ), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.

Compactifications of locally compact groups and closed subgroups

Let G be a locally compact group with closed normal subgroup N such that G/N is compact. In this paper, we construct various semigroup compactifications of G from compactifications of N of the same type. This enables us to obtain specific information about the structure of the compactification of G from the structure of the compactification of N. Our results seem to be interesting and new even when G is the additive group of real numbers and N is the integers. Applications and other examples are given.

On distal and equicontinuous compact right topological groups

W. Ruppert has studied, and given examples of, compact left topological groups for which the left translation flow (Aa, G) is equicontinuous. Recently, we considered an anal- ogous distal condition that applies to the groups of dynamical type; for these the topological centre is dense, so the translation flow is equicontinuous only in the trivial case when G is topological. In the present paper, we continue this work, exhibiting new characterizations of equicontinuity and distality of (Aa, G); we also discuss examples and study some related matters.

Left mean-ergodicity, fixed points, and invariant means☆

Abstract Recently Lau [15] generalized a result of Yeadon [25]. In the present paper we generalize Yeadons result in another direction recasting it as a theorem of ergodic type. We call the notion of ergodicity required left mean-ergodicity and show how it relates to the mean-ergodicity of Nagel [21]. Connections with the existence of invariant means on spaces of continuous functions on semitopological semigroups S are made, connections concerning, among other things, a fixed point theorem of Mitchell [20] and Schwartzs property P of W∗-algebras [22]. For example, if M (S) is a certain subspace of C(S) (which was considered by Mitchell and is of almost periodic type, i.e., the right translates of a member of M (S) satisfy a compactness condition), then the assumption that M (S) has a left invariant mean is equivalent to the assumption that every representation of S of a certain kind by operators on a linear topological space X is left mean-ergodic. An analog involving the existence of a (left and right) invariant mean on M (S) is given, and we show our methods restrict in the Banach space setting to give short direct proofs of some results in [4], results involving the existence of an invariant mean on the weakly almost periodic functions on S or on the almost periodic functions on S. An ergodic theorem of Lloyd [16] is generalized, and a number of examples are presented.

Almost periodic compactifications of group extensions

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Multiplicative polynomials and Fermat's little theorem for non-primes

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