The group of characters of a pseudocompact locally compact semitopological semigroup

1. The semigroup analogue of the Pontrjagin duality theorem was first studied in [1]. In that paper, it was shown that a necessary and sufficient condition for duality in discrete abelian semigroups is that the semigroup be a union of groups and have an identity element. Such semigroups we shall call inverse semigroups. For compact abelian topological semigroups it was shown in [1] that the separation of points by semicharacters is a sufficient condition for duality in an inverse semigroup with identity. In [2] it was shown that, in any topological abelian semigroup, a necessary condition for duality is that the semigroup be an inverse semigroup with identity and continuous inversion. In this paper we obtain necessary and sufficient conditions that semicharacters separate points in a topological abelian inverse semigroup with identity which is compact, or locally compact with continuous inversion. In the compact case we obtain the same result as has been given by Sneperman [5], using different methods. 1.1. DEFINITION. An abelian semigroup is a nonempty set together with a map m: (x, y)-*xy on SXS to S, such that x(yz) = (xy)z and xy =yx for all x, y and z in S. If is a Hausdorff topological space and the mapping m is continuous, is called a topological abelian semigroup. 1.2. DEFINITION. A semicharacter X of a topological abelian semigroup is a bounded, continuous, complex-valued function on S, not identically zero, satisfying x(xy) =x(x)x(y) for all x and y in S. We denote the set of semicharacters of by SA. We endow SA with the compact open topology. The following facts aretobefoundin [1], [3]or [4]. 1.3. If has an identity element, S^ becomes a topological abelian semigroup, when endowed with the operation of pointwise multiplication. 1.4. If has an identity element, and is discrete, S^ is a compact abelian semigroup [1, 3.1 ]. 1.5. We call an abelian inverse semigroup if is an abelian semigroup which is a union of groups. If is a topological, abelian inverse semigroup with an identity element then S is of the same type. Further, if is compact then S^ is discrete [1, 6.11.

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.

The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.

We study the structure of inverse primitive pseudocompact semitopological and topologi- cal semigroups. We nd conditions when the maximal subgroup of an inverse primitive pseudocompact semitopological semigroupS is a closed subset ofS and describe the topological structure of such semireg- ular semitopological semigroups. Later we describe the structure of pseudocompact topological Brandt 0 -extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse pseudocompact topological semigroup is continuous. Also an analogue of Comfort{Ross Theorem proved for such semi- groups: a Tychono product of an arbitrary family of primitive inverse semiregular pseudocompact semitopological semigroups with closed maximal subgroups is pseudocompact. We describe the structure of the Stone- Cech compactication of a Hausdor primitive inverse countably compact semitopological semigroup S such that every maximal subgroup of S is a topological group.

This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull Ω ( S ) \Omega (S) is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact-open topology in this case), then Ω ( S ) \Omega (S) is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divisibility are presented. Techniques for determining the translational hull of certain types of topological semigroups, along with numerous examples, are set forth in the final section.

0. Introduction. A semigroup S is (uniquely) divisible if, for each x E S, and each positive integer n, there exists (a unique) y E S such that yn = x. In the unique case we write y=X1ln. Uniquely divisible commutative semigroups, referred to in the sequel as UDC semigroups, have been characterized in [7]. Compact topological semigroups satisfying this hypothesis have been studied in [6], [11], [12], and [13]. Material of a related nature occurs in [15] and [17]. In [6], the authors showed that if S is a finite-dimensional compact UDC semigroup in which the set of idempotents is totally disconnected, then there exist sufficiently many continuous homomorphisms (semicharacters) of S into the complex unit disk to separate points. This usage of the complex disk as a range space is in line with the classical philosophy, group is to the circle group as Abelian semigroup is to the complex unit disk semigroup. A large amount of work has been done on the investigation of this analogy; in particular, see [9] for a comprehensive survey of the algebraic results in this direction. For the case of Abelian topological semigroups and continuous semicharacters, see [16] and [23]. However, it is clear that the idempotent structure of the complex disk makes it unsuitable as a range space for continuous homomorphisms on general Abelian topological semigroups. In particular, if S is a connected topological semilattice, then any continuous homomorphism of S into the complex disk must be trivial. This deficiency is well known, and reasonable substitutes for the complex disk have been sought for some time. If each maximal group of S is trivial, then one of the most appealing replacements for the disk is some form of thread -a semigroup on a space homeomorphic to the unit interval in which one endpoint acts as an identity and the other as a zero. The complete structure of threads is given in [21], and questions concerning their suitability as range spaces for continuous semicharacters are raised therein. The basic building blocks for threads are U, the interval [0, 1] under real number multiplication; M, the interval [0, 1] under multiplication xy =min {x, y}; and C, the interval [1/2, 1] under multiplication x y = max {1 /2, xy}, where xy represents the ordinary real number product of x and y. The semigroup U has been used successfully as a range space for a certain class of compact UDC semigroups in [6]. The semigroup M is a very logical range space for the category of compact topological semilattices; the problem of whether every

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.

The structure of idempotent probability measures on compact topological semigroups is well known (see, for example, [2], [41, [7] and [9]). However, the statement in [8] that the methods of [7] can be used to obtain identical results when the semigroup is only semitopological (i.e. the multiplication is separately rather than jointly continuous) is misleading, since the minimal ideal of a compact semitopological semigroup may not be closed [1, Chapter IV, Example 7.1]. In this note, it will be indicated how the convolution decomposition of a probability idempotent may be achieved in the semitopological case. Let S be a compact semitopological semigroup. For each continuous function f on S and each pair x, y of elements of S, we write xf(y) =f(xy) and f.(y) =f (yx). If v is a Radon measure on S we put K(f) = ff (x)dv(x), and define f (x) = v(fT) and f,(x) =v(Jf); the functions Jf and fv are continuous (see [3]). If X is any other Radon measure on S, the mapping (x, y)-*f(xy) is measurable with respect to the product measure X?v (see [5] or [6]), and (using the Fubini Theorem) we may define the convolution product X * v by writing

The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if B is the bicyclic semigroup and if B is contained in any semitopological semigroup, the relative topology on B is discrete. We show that if F is an inverse semigroup generated by an element and its inverse, and F is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if F is any one-parameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on F is jointly continuous if and only if the inversion is continuous on F, and we describe F ¯ \bar F in that case. We also show that if { J t } \{ {J_t}\} is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then { J t } ∪ { J t ∗ } \{ {J_t}\} \cup \{ J_t^\ast \} generates a one-parameter inverse semigroup T with J t − 1 = J t ∗ J_t^{ - 1} = J_t^\ast if and only if { J t } \{ {J_t}\} is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup [Formula: see text]. The set of all quasi-multipliers on [Formula: see text] is denoted by [Formula: see text]. First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if [Formula: see text] is a topological semigroup such that [Formula: see text] is pseudocompact, then [Formula: see text] can be regarded as a subset of [Formula: see text] Moreover, with an extra condition we describe [Formula: see text] as a quotient subsemigroup of [Formula: see text] Finally, we investigate quasi-multipliers on topological semigroups, its relationship with multipliers and give some concrete examples.

or what is equivalent, if for k in the complement of a set of density 0, m(E r) T kF) -+ m(E)m(F), for each pair of measurable sets E, F. The so-called mixing theorem of ergodic theory (see Halmos [5, p. 39] ) establishes the equivalence of (1.1) and each of the following two conditions: first, that the unitary operator UT on L2 (F, S, m) induced by T has continuous spectrum on the subspace of L2 orthogonal to the constant functions; and second, that the Cartesian square T x T on the product space (F x F, S x S, n x m) isergodic. This theorem appeared early in the modern development of ergodic theory, in works of von Neumann and Hopf, and unlike most of these early results, it appears to have resisted subsequent generalization. For example, no variant of the theorem seems to be known in case the measure-preserving transformation T is noninvertible. In this situation the induced operator UT is, in general, a nonunitary isometry, to which ordinary spectral theory does not apply. In this note we prove an abstract mixing theorem in a setting considerably more general than the above. In place of the semigroup of non-negative integers we consider an arbitrary topological semigroup G which admits both a leftinvariant and a right-invariant Banach mean. This includes all abelian or compact topological semigroups, and an extensive class of discrete semigroups (see Day [1]). And, in place of the operator UT, we consider an arbitrary weakly continuous isometric representation U. of G on a complex Hiubert space. The operation lim (1 / n) Sk-IiS supplanted by almost convergence, a notion introduced by G. Lorentz [6], and in turn, the condition that UT have continuous spectrum is supplanted by the condition that the representation U, have no finite-dimensional subrepresentations. The abstract mixing theorem (proved in ?3) specializes in the measure-theoretic case to a direct generalization of the classical mixing theorem (see ?4). Our proof is quite elementary, consisting of a refinement for amenable semigroups of the traditional Peter-Weyl theory.

Let X, Y, Z be topological spaces. A function F :X × Y → Z is called jointly continuous if it is continuous from X × Y with the product topology to Z . It is said to be separately continuous if x 7→ F (x, y):X → Z is continuous for each y ∈ Y and y 7→ F (x, y):Y → Z is continuous for each x ∈ X . A semitopological semigroup is a semigroup S endowed with a topology such that the multiplication function is separately continuous, or equivalently, all left and right translations are continuous. In the case the semigroup is actually a group, we call it a semitopological group (it is often required that the inversion function be continuous also, but we omit this assumption). There is considerable literature (see [5] for a survey and bibliography) devoted to the problem of embedding a topological semigroup (a semigroup with jointly continuous multiplication) into a topological group, but the semitopological version appears to have received scant attention. On the other hand there are good reasons for considering this case, among them the fact that a number of general conditions exist for concluding that a semitopological group is actually a topological one. We further find that the semitopological setting gives much more straightforward statements and proofs of key results. The ideas of this paper parallel in many aspects those contained in the first part of [1]. Our Corollary 2.7 is essentially Theorem 2.1 of [1] extended from right reversible semigroups to general semigroups embedded in a group. Results paralleling most of the results of this paper can be found in Chapter VII of [4], except that we relax the hypothesis assumed on the semigroup S to only assuming translations are open mappings (a stronger condition on the semigroups S is assumed in [4] to guarantee continuity of inversion in the containing group). In addition, our methods here are much quicker and more direct. The algebraic problem of giving necessary and sufficient conditions for group embeddability of a semigroup is a delicate one, although cancellativity is an obvious necessary condition. We bypass the algebraic problem by considering only semigroups that are (algebraically) group embeddable. If a semigroup S embeds in a group G , then there is a smallest subgroup in G containing S and we assume that our embeddings are always readjusted at the codomain level so that S generates (as a group) G .

In recent years, the Stone-Čech compactification of certain semigroups (e.g. discrete semigroups) has been an interesting semigroup compactification (i.e. a compact right semitopological semigroup which contains a dense continuous homomorphic image of the given semigroup) to study, because an Arens-type product can be introduced. If G is a non-compact and non-discrete locally compact abelian group, then it is not possible to introduce such a product into the Stone-Čech compactification βG of G (see [1]). However, let UC(G) be the Banach algebra of bounded uniformly continuous complex functions on G, and let UG be the spectrum of UC(G) with the Gelfand topology. If f∈ UC(G), then the functions f and fy defined on G byare also in UC(G).

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$.

In probability theory, often in connection with problems on weak convergence, and also in other contexts, convolution equations of the form $$\sigma *\mu =\mu $$ come up. Many years ago, Choqet and Deny (C. R. Acad. Sci. Paris 250 (1960) 799–801) studied these equations in locally compact abelian groups. Later, Szekely and Zeng (J. Theoret. Probab. 3(2) (1990) 361–365) studied these equations in abelian semigroups. Like in [2], the results in [7] are also complete. Thus, these equations are studied here for the first time on non-compact non-abelian semigroups. Our main results are Theorems 3.1 and 3.3 in section 3. They are new results as far as we know, and also the best possible under a minor condition. All semigroups in this paper are, unless otherwise mentioned, locally compact Hausdorff second countable topological semigroups. Theorems 3.1 and 3.3 hold for these semigroups. Local compactness may not be necessary when all measures appearing in this context are assumed to be regular.