Compact Semitopological Semigroup Research Articles

On topological McAlister semigroups

In this paper we investigate algebraic and topological properties of McAlister semigroups. We show that for a non-zero cardinal λ the group of automorphisms of the McAlister semigroup Mλ is isomorphic to the direct product Sym(λ)×Z2, where Sym(λ) is the group of permutations of λ. McAlister semigroups admit a unique compact Hausdorff semigroup topology. Each non-zero element of a Hausdorff semitopological McAlister semigroup is isolated. It follows that the free inverse semigroup over a singleton admits only the discrete Hausdorff shift-continuous topology. We prove that a Hausdorff locally compact semitopological semigroup M1 is either compact or discrete. However, this dichotomy does not hold for the semigroup M2. Moreover, M2 admits continuum many different Hausdorff locally compact inverse semigroup topologies.

On a locally compact monoid of cofinite partial isometries of ℕ with adjoined zero

Abstract Let 𝒞ℕ be a monoid which is generated by the partial shift α : n↦n +1 of the set of positive integers ℕ and its inverse partial shift β : n + 1 ↦n. In this paper we prove that if S is a submonoid of the monoid Iℕ∞ of all partial cofinite isometries of positive integers which contains Cscr;ℕ as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

Two open problems of Day and Wong on left thick subsets and left amenability

This paper answers two open problems raised by Mahlon M. Day [4] and James C.S. Wong [17]: Does uniformly topological left amenability imply the existence of left translation continuous measures on a locally compact semitopological semigroup? Let T be a locally compact Borel subsemigroup of a locally compact semitopological semigroup S. Is the existence of a topological T-invariant mean M on S with M(χT)>0 enough to imply the topological left amenability of T? We give a negative answer to the first question by providing a counterexample and a positive proof to the second. This example can also be used to show that the property (α) provided in Gerard L. Sleijpen [14] can not be removed in one of his results.

On feebly compact inverse primitive (semi)topological semigroups

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.

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Cech compactification βS is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient \(\ensuremath {\mathcal {C}_{b}}(S)/\ensuremath {\mathcal {LMC}}(S)\) contains a linear isometric copy of l∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.

SPECTRUM OF THE FOURIER-STIELTJES ALGEBRA OF A SEMIGROUP

For a unital foundation topological -semigroup S whose rep- resentations separate points of S, we show that the spectrum of the Fourier-Stieltjes algebra B(S) is a compact semitopological semigroup. We also calculate B(S) for several examples of S.

On the Space of Weakly Almost Periodic Functionals and Its Amenability

Let S be a locally compact semitopological semigroup with measure algebra M(S), M0(S) the set of all probability measures in M(S) and WF(S) the space of weakly almost periodic functionals on M(S)*. Assuming that M0(S) has the semiright invariant isometry property, it is shown that WF(S) has a topological left invariant mean (TLIM) whenever the center of M0(S) is nonempty; in particular if either the center of S is nonempty or S has a left identity, then WF(S) has a TLIM. Finally if, for each μ ∈ M0(S), the mapping v → v * μ of M0(S) into itself is surjective and the center of M0(S) is nonempty, then WF(S) has a TLIM. We also generalize some results from discrete case to topological one.

Banach-valued probability measures

We introduce and study the notion of Banach-valued probability measures on a compact semitopological semigroup. In particular, we prove that these measures are nontrivial idempotents and convolution is separately continuous. We give an example of a Banach-valued measure where the support may not be simple; though for any idempotent measure the support is a closed simple subsemigroup.

A Compact Monothetic Semitopological Semigroup Whose Set of Idempotents Is Not Closed

For each monothetic unitary group without proper co-compact subgroups, we show the existence of a compact semitopological semigroup (affine) compactification in which the set of idempotents is not closed answering a question of Berglund.

The uniform compactification of a locally compact abelian group

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).

Vector-valued probability measures on semigroups

The problem of defining vector-valued probability measures on a compact semitopological semigroup S is considered in this paper. The set PW( S, A) of all such A-valued measures, analogous to the scalar-valued measures, is a compact semitopological semigroup with respect to the convolution operation and the weak operator topology. Supports of idempotents and the limits of averaged convolution sequences in PW( S, A) are determined.

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.

Weakly compact operators and multipliers on C0(S, A)

The problem of defining a convolution of operator-valued measures defined on a locally compact semitopological semigroup which represent weakly compact operators is considered in this paper. Convolution of the corresponding weakly compact operators is then defined. A characterization of multipliers on function spaces is given in terms of their representing measures.

Conditionally compact semitopological one-parameter inverse semigroups of partial isometries

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.

On supports of semigroups of measures

Let S denote a compact semitopological semigroup (i.e. the multiplication is separately continuous) and P(S) the set of probability measures on S. Then P(S) is a compact semitopological semigroup under convolution and the weak * topology (4). Let Γ be a subsemigroup of P(S) and where supp μ is the support of μ ∈P(S). In the case in which S is commutative it was shown by Glicksberg in (4) that S(Γ) is an algebraic group in S if Γ is an algebraic group. For a general semigroup S, Pym (7) considered Γ = {η}, η being an idempotent, and established that S(Γ) is a topologically simple subsemigroup of S, i.e. every ideal of S(Γ) is dense in S(Γ). In this note we prove that if Γ is a simple subsemigroup of P(S) (a semigroup is simple if it contains no proper ideal) which contains an idempotent then S(Γ) is a topologically simple subsemigroup of S. We also give an example to show that our conclusion (hence also Pym's) is best possible in the sense that S(Γ) is not simple in general

Limit Measures on Compact Semitopological Semigroups.

Limit Measures on Compact Semitopological Semigroups.

On central primitive idempotent measures

Let S be a compact semitopological semigroup and let P(S) be the convolution semigroup of probability measures on S. An idempotent measure μ in P(S) is defined to be primitive if and only idempotent measures in μP(S)μ are μ and the zero element m of P(S). In a previous paper [2] we give some characterization of primitive idempotent measures on S. Let Π(P(S)) be the set of primitive idempotents in P(S) and let Πc be the set of central primitive idempotents in P(S). It is shown in [1] that Π(P(S)) is neither an ideal nor even a subsemigroup of P(S) in general. The purpose of this paper is to investigate the structure of Πc.

Locally compact subgroups of the spectrum of the measure algebra III

The maximal ideal space of the measure algebra of a locally compact abelian (LCA) group has the structure of a compact commutative semitopological semigroup (separately continuous multiplication). Idempotents in the semigroup correspond to certain algebraic projections on the measure algebra. In this paper we study the maximal groups about certain idempotents.

Primitive idempotent measures on compact semitopological semigroups

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.