© Hindawi Publishing Corp. REDUCTIVE COMPACTIFICATIONS OF SEMITOPOLOGICAL SEMIGROUPS

We consider the enveloping semigroup of a flow generated by the 
action of a semitopological semigroup on any of its semigroup 
compactifications and explore the possibility 
of its being one of the known semigroup compactifications again. 
In this way, we introduce the notion of E-algebra, and show that 
this notion is closely related to the reductivity of the semigroup 
compactification involved. Moreover, the structure of 
the universal Eℱ-compactification is also given.


Introduction.
A semigroup S is called right reductive if a = b for each a, b ∈ S, since at = bt for every t ∈ S. For example, all right cancellative semigroups and semigroups with a right identity are right reductive.
From now on, S will be a semitopological semigroup, unless otherwise is stipulated.By a semigroup compactification of S we mean a pair (ψ, X), where X is a compact Hausdorff right topological semigroup, and ψ : S → X is a continuous homomorphism with dense image such that, for each s ∈ S, the mapping x→ψ(s)x : X→X is continuous.The C * -algebra of all bounded complex-valued continuous functions on S will be denoted by Ꮿ(S).For Ꮿ(S), the left and right translations, L s and R t , are defined for each s, t ∈ S by (L s f ) where S Ᏺ is the spectrum of Ᏺ).Then the product of µ, ν ∈ S Ᏺ can be defined by µν = µ • T ν and the Gelfand topology on S Ᏺ makes ( , S Ᏺ ) a semigroup compactification (called the Ᏺ-compactification) of S, where : S→S Ᏺ is the evaluation mapping.Some m-admissible subalgebras of Ꮿ(S), that we will need, are left multiplicatively continuous functions ᏸᏹᏯ, distal functions Ᏸ, minimal distal functions ᏹᏰ, and strongly distal functions Ᏸ.We also write Ᏻᏼ for ᏹᏰ ∩ Ᏸ; and we define ᏸᐆ := {f ∈ Ꮿ(S); f (st) = f (s) for all s, t ∈ S}.For a discussion of the universal property of the corresponding compactifications of these function algebras see [1,2].

Reductive compactifications and E-algebras.
Let (ψ, X) be a compactification of S, then the mapping σ : S × X→X, defined by σ (s,x) = ψ(s)x, is separately continuous and so (S,X,σ ) is a flow.If Σ X denotes the enveloping semigroup of the flow (S,X,σ ) (i.e., the pointwise closure of semigroup {σ (s,•) : s ∈ S} in X X ) and the mapping σ X : S→Σ X is defined by σ X (s) = σ (s,•) for all s ∈ S, then (σ X , Σ X ) is a compactification of S (see [1,Proposition 1.6.5]).
One can easily verify that Σ X = {λ x : x ∈ X}, where λ x (y) = xy for each y ∈ X.If we define the mapping θ : X → Σ X by θ(x) = λ x , then θ is a continuous homomorphism with the property that θ For example, the ᏹᏰ-, Ᏻᏼ-, and ᏸᐆ-compactifications are reductive.
An m-admissible subalgebra Ᏺ of Ꮿ(S) is called an E-algebra if there is a compactification (ψ, X) such that (σ X , Σ X ) ( , S Ᏺ ).In this setting (ψ, X) is called an EᏲ-compactification of S. Trivially for every reductive compactification (ψ, X), ψ * (Ꮿ(X)) is an E-algebra.But the converse is not, in general, true.For instance, for any compactification (ψ, X), σ * X (Ꮿ(Σ X )) is an E-algebra; however, it is possible that Σ X would be nonreductive, as the next example shows.
Example 2.2.Let S = {a, b, c, d} be the semigroup with the following multiplication table: Then for the identity compactification (i, X) of S, Σ X is not right reductive; in fact, λ a ≠ λ b , however, λ at = λ bt for every t ∈ S.
Corollary 2.4.Let sS (or Ss) be dense in S, for some s ∈ S, then for every compactification (ψ, X) of S, it follows that X 2 = X and so (σ X , Σ X ) is reductive.Now, we are going to construct the universal EᏲ-compactification of S. For this end we need the following lemma.Lemma 2.5.Let Ᏺ be an m-admissible subalgebra of Ꮿ(S).Then T ν f ∈ σ * S Ᏺ (Ꮿ(Σ S Ᏺ )) for all f ∈ Ᏺ and ν ∈ S ᏸᏹᏯ .
Proposition 2.6.Let Ᏺ be an E-algebra.Then is an m-admissible subalgebra of Ꮿ(S) and ( , S GᏲ ) is the universal EᏲ-compactification of S.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.