Periodic Compactification Research Articles

On group topologies and idempotents in weak almost periodic compactifications

On group topologies and idempotents in weak almost periodic compactifications

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT5, be the translate ofT bys inS defined byT5(x)=(Tx)5. We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T5. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μs, where μs denotes the unique vector measure corresponding to the operatorT5.

A Seminar on almost periodic compactifications continuity, and compact semigroups

A Seminar on almost periodic compactifications continuity, and compact semigroups

Filters and the Weak Almost Periodic Compactification of a Discrete Semigroup

The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.

Filters and the weak almost periodic compactification of a discrete semigroup

The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.

Semigroup Compactifications of Direct and Semidirect Products

AbstractA classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

Semidirect Product Compactifications

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

An amenability property of algebras of functions on semidirect products of semigroups

Let S1 and S2 be semitopological semigroups, S1 τ S2 a semidirect product. An amenability property is established for algebras of functions on S1 τ S2. This result is used to decompose the kernel of the weakly almost periodic compactification of S1 τ S2 into a semidirect product.

Semigroup compactifications of semidirect products

Let S S and T T be semigroups, S\text {\textcircled {𝜏 }} T a semidirect product, and F F a C ∗ {C^ \ast } -algebra of bounded, complex-valued functions on S\text {\textcircled {𝜏 }} T. Necessary and sufficient conditions are given for the F F -compactification of S\text {\textcircled {𝜏 }} T to be expressible as a semidirect product of compactifications of S S and T T . This result is used to show that the strongly almost periodic compactification of S\text {\textcircled {𝜏 }} T is a semidirect product and that, in certain general cases, the analogous statement holds for the almost periodic compactification and the left uniformly continuous compactification of S\text {\textcircled {𝜏 }} T. Applications are made to wreath products.

Almost periodic compactifications of direct and semidirect products

Almost periodic compactifications of direct and semidirect products

Almost periodic functions on semidirect products of transformation semigroups

The notion of semidirect product of two transformation semigroups is introduced, and its space of almost periodic functions is expressed as a tensor product. The general techniques developed are applied to the special case of a semidirect product S © T of two semigroups. As a consequence new results are obtained on the characterization of the almost periodic compactification of S © Γ as a semidirect product of compact semigroups. A related result is the splitting of the enveloping semigroup of a semidirect product of certain flows into a semidirect product of enveloping semigroups. 0. Introduction. Let S and T be semitopological semigroups and S (?) Γ a semidirect product of S and T. In an earlier paper [10] we showed that, under certain conditions, the almost periodic (a.p.) compactification (S © T) r of S © T is a semidirect product of the a.p. compactification of T and a certain compact topological semigroup containing a dense homomorphic image of S. A simple corollary of this result is that the space of a.p. functions on S © T is a tensor product of the space of a.p. functions on T and a subspace of a.p. functions on S. In this paper we introduce the notion of semidirect product of transformation semigroups and determine exactly when its space of a.p. functions may be expressed as a tensor product in analogy with the semigroup case described above. Cast in this general setting the problem of characterizin g the space of a.p. functions on a semidirect product of semigroups becomes clear, and the techniques developed lead to elegant necessary and sufficient conditions for (S © TY to be a semidirect product. As a consequence we are able to show that (S © T)' is a semidirect product for all semitopological semigroups S with identity and all semitopological groups T, thus generalizing results of [10, 11, 12]. The same conclusion holds if T merely contains a dense subgroup. In a similar vein, but using different techniques, we show that in a wide variety of cases the enveloping semigroup of the semidirect product of two equicontinuous flows is (canonically isomorphic to) a semi- direct product of the original enveloping semigroups.

Algebras of functions on semitopological left-groups

We consider various algebras of functions on a semitopological left-group S = X × G S = X \times G , the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products S × T S \times T of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.

On the structure of the Fourier-Stieltjes algebra

If G is a locally compact group, denote its Fourier-Stieltjes algebra by B(G) and its Fourier algebra by Λ(G). If G is compact, then B(G) = Λ(G) and σ(B(G)), the spectrum o! B(G), is G. If G is not compact then σ(B(G)) contains partial isometries and projections different from e, the identity of G. More generally, σ(B(G)) is closed under operations that commute with representing and the taking of tensor products. It is shown that σ(B(G)) contains a smallest positive element, zF; and that g G G Cσ(B(G))h+z Fg G σ(B(G))zF is an epimorphism of G into G, the almost periodic compactification of G. A structure theorem is given for the closed, bi-translation, invariant subspaces of B(G). In so doing we introduce the concepts of inverse Fourier transform localized at TΓ, and the standardization of TΓ, where TΓ is a continuous, unitary representation of G.

The weakly almost periodic compactification: Another approach

The weakly almost periodic compactification: Another approach

3.—Some Structure Semigroup Results for Arens-Singer Semigroups

SynopsisIn this paper we discuss the structure semigroup of the L1-algebra of an Arens-Singer semigroup. Arising from this study we provide a complete and rather unexpected description of a nontrivial structure semigroup. We then link the above ideas with that of the almost periodic compactification of the semigroup.

Applications of almost periodic compactifications

Applications of almost periodic compactifications

A remark on some almost periodic compactifications.

A remark on some almost periodic compactifications.