Compact Semigroup Research Articles

Density and invariant means in left amenable locally compact topological groups

Amenable groups have a close relationship with various mathematical concepts. In this paper we consider the correlation between the amenability and semigroup compactification of a locally compact group. It is observed that in the case that G is an amenable group, there are some wonderful properties for GLUC. Motivated by the definition provided by N. Hindman and D. Strauss, we defined the set LIM0(G) and other sets such as Φ(G) and Δ⁎(G) to achieve the desired results and we considered the correlation between the amenability and semigroup compactification of a locally compact group.

Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points

The spectrum of an admissible subalgebra A(G) of LUC(G)(G), the algebra of right uniformly continuous functions on a locally compact group G, constitutes a semigroup compactification GA of G. In this paper we analyze the algebraic behaviour of those points of GA that lie in the closure of A(G)-sets, sets whose characteristic function can be approximated by functions in A(G). This analysis provides a common ground for far reaching generalizations of Veech's property (the action of G on GLUC(G) is free) and Pym's Local Structure Theorem. This approach is linked to the concept of translation-compact set, recently developed by the authors, and leads to characterizations of strongly prime points in GA, points that do not belong to the closure of G⁎G⁎, where G⁎=GA∖G. All these results will be applied to show that, in many of the most important algebras, left invariant means of A(G) (when such means are present) are supported in the closure of G⁎G⁎.

Asymptotic Stability of Neutral Impulsive Stochastic Partial Differential Equation of Sobolev Type with Poisson Jumps

This paper is concerned with the p-th asymptotical stability of the mild solution of an impulsive neutral stochastic integro-differential equation of Sobolev type involving Poisson jumps and non-instantaneous impulses. By utilizing Leray–Schauder fixed point theorem and compact semigroup, a set of sufficient conditions establishing the asymptotical stability of the mild solution to stochastic system in the p-moment is derived. An example is also considered for illustrating abstract results.

Controllability of Second-Order Impulsive Nonlocal Cauchy Problem Via Measure of Noncompactness

In this article, we consider a class of second-order impulsive evolution differential equations with nonlocal conditions. This article deals with the nonlocal controllability for a class of second-order evolution impulsive control systems. We prove some sufficient conditions for controllability using the measure of noncompactness and Monch fixed point theorem. Very particularly we do not assume that the evolution system generates a compact semigroup. Finally, an example is given to represent the obtained theory.

Strict topology on locally compact semigroups

Let S be a locally compact Hausdorff semigroup, and \(\mathfrak {A}\) a solid subalgebra of measure algebra M(S). In this paper, among other results, we find necessary and sufficient conditions on S that implies \(\mathfrak {A}\) is a semi-topological or a topological algebra with respect to the strict topology on M(S). Applications to discrete semigroups, Brandt semigroups and Clifford semigroups are given. An example establishes negatively the open question of Maghsoudi (Semigroup Forum 86:133–139, 2012). Also, we give a correct proof of Proposition 2.1 of Maghsoudi (2012).

Infinite compact sets of idempotents in $$\beta S$$ β S

We show that if S is a countably infinite right cancellative semigroup and T is an infinite compact set of idempotents in the Stone–Cech compactification \(\beta S\) of S, then T contains an infinite compact left zero semigroup.

Fixed point compactifications

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that \(ap=pa=p\) for all \(p\in S\setminus G\) and \(a\in G\). Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted \(\theta G\). We show that if G is \(\sigma \)-compact, then \(\theta G\setminus G\) contains a copy of \(\beta \omega \setminus \omega \). In contrast, if G is not \(\sigma \)-compact, then \(\theta G\) is the one-point compactification.

More on the locally convex space $(M(X),\beta(X))$ of a locally compact Hausdorff space $X$

In the previous paper [12] we introduced the definition of the strict topology $\beta(X)$ on the measure space $M(X)$ for a locally compact Hausdorff space $X$. In this paper, we consider on $M(X)$ the topology $\beta(X)$ and we show that $\beta(X)$ is the weak topology under all left multipliers induced by a function space on $M(X)$. We then show that $\beta(X)$ can be considered as a mixed topology. This result is not only of interest in its own right, but also it paves the way to prove that $(M(X),\beta(X))$ is a Mazur space and the locally convex space $(M(S),\beta(S))$, equipped with the convolution multiplication is a complete semitopological algebra, for a wide class of locally compact semigroups $S$.

The algebraic structure of quantum partial isometries

The partial isometries of [Formula: see text] form compact semigroups [Formula: see text]. We discuss here the liberation question for these semigroups, and for their discrete versions [Formula: see text]. Our main results concern the construction of half-liberations [Formula: see text] and of liberations [Formula: see text]. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.

Extremal Positive Maps on M3(ℂ) and Idempotent Matrices

A new method of analysing positive bistochastic maps on the algebra of complex matrices [Formula: see text] has been proposed. By identifying the set of such maps with a convex set of linear operators on [Formula: see text], one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been shown that all positive bistochastic maps, extremal in the set of all positive maps of [Formula: see text], that are not Jordan isomorphisms of [Formula: see text] are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one-dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.

Analyticity and compactness of semigroups of composition operators

This paper provides a complete characterisation of quasicontractive groups and analytic C0-semigroups on Hardy and Dirichlet space on the unit disc with a prescribed generator of the form Af=Gf′. In the analytic case we also give a complete characterisation of immediately compact semigroups. When the analyticity fails, we obtain sufficient conditions for compactness and membership in the trace class. Finally, we analyse the case where the unit disc is replaced by the right-half plane, where the results are drastically different.

Quantum Eberlein compactifications and invariant means

We propose a definition of a � algebra, which is a weak form of a C � -bialgebra with a sort of unitary generator. Our definition is motivated to ensure that commutative examples arise exactly from semigroups of contractions on a Hilbert space, as extensively studied recently by Spronk and Stokke. The terminology arises as the Eberlein algebra, the uniform closure of the Fourier-Stieltjes algebra B(G), has character space G E , which is the semigroup compactification given by considering all semigroups of contractions on a Hilbert space which contain a dense homomorphic image of G. We carry out a similar construction for locally compact quantum groups, leading to a maximal C � - Eberlein compactification. We show that C � -Eberlein algebras always admit invariant means, and we apply this to prove various splitting results, showing how the C � -Eberlein compactification splits as the quantum Bohr compactification and elements which are annihilated by the mean. This holds for matrix coefficients, but for Kac algebras, we show it also holds at the algebra level, generalising (in a semigroup-free way) results of Godement. 2010 Mathematics Subject Classification: Primary 43A07, 43A60, 46L89, 47D03; Secondary 22D25, 43A30, 43A65, 47L25

On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces

The present paper studies the initial value problem of stochastic evolution equations with compact semigroup in real separable Hilbert spaces. The existence of saturated mild solution and global mild solution is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate that our results are valuable.

RETRACTED ARTICLE: Function spaces and compact semigroups associated with oids

RETRACTED ARTICLE: Function spaces and compact semigroups associated with oids

A Parametrization of the Irreducible Representations of a Compact Inverse Semigroup

The aim in this article is to provide a parametrization of the finite dimensional irreducible representations of a compact inverse semigroup in terms of the irreducible representations of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new, and more conceptual, proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Moreover, we also prove that every norm continuous irreducible *-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

West semigroups as compactifications of locally compact abelian groups

In this paper, we will identify certain subsemigroups of the unit ball of \(L^{\infty }[0,1]\) as semitopological compactifications of locally compact abelian groups, using an idea of West (Proc R Ir Acad Sect A 67:27–37, 1968). Our result has been known for the additive group of integers since Bouziad et al. (Semigr Forum 62(1):98–102, 2001). We will construct a semitopological semigroup compactification for each locally compact abelian group G, depending on the algebraic properties of G. These compact semigroups can be realized as quotients of both the Eberlein compactification \(G^e\), and the weakly almost periodic compactification, \(G^w\), of G. The concrete structure of these compact quotients allows us to gain insight into known results by Brown (Bull Lond Math Soc 4:43–46, 1972) and Brown and Moran (Proc Lond Math Soc 22(3):203–216, 1971) and by Bordbar and Pym (Math Proc Camb Philos Soc 124(3):421–449, 1998), where for the groups \(G=\mathbb {Z}\) and \(G=\mathbb {Z}_q^{\infty }\), it is proved that \(G^e\) and \(G^w\) contain uncountably many idempotents and the set of idempotents is not closed.

Existence results in the $$\alpha $$ α -norm for neutral partial functional integro-differential equations with infinite delay

In this paper we propose to study, in the \(\alpha \)-norm, a class of neutral partial functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic and compact semigroup and the nonlinear part of the system involve spatial derivatives. At the end, an example is provided to illustrate the application of the obtained results.

FILTERS AND THE WEAKLY ALMOST PERIODIC COMPACTIFICATION OF A SEMITOPOLOGICAL SEMIGROUP

Let S be a semitopological semigroup. The wap− compactification of semigroup S, is a compact semitopological semi- group with certain universal properties relative to the original semi- group, and the Lmc− compactification of semigroup S is a univer- sal semigroup compactification of S, which are denoted by S wap and S Lmc respectively. In this paper, an internal construction of the wap−compactification of a semitopological semigroup is con- structed as a space of z−filters. Also we obtain the cardinality of S wap and show that if S wap is the one point compactification then (S Lmc − S) ∗ S Lmc is dense in S Lmc − S.

On the spectra of [formula omitted]-algebras of slowly oscillating functions on locally compact groups

We present a study of C⁎-algebras SO(φ) of slowly oscillating functions in the direction of filters φ on a locally compact topological group G. We show that SO(φ) is an m-admissible subalgebra of C(G) if and only if the closure of the filter φ in the LUC-compactification GLUC of G is an ideal of GLUC and that the semigroup compactification of G determined by SO(φ) always contains right zero elements. Using this, we characterize a new interesting C⁎-algebra of bounded continuous functions on G. The spectrum of this C⁎-algebra determines the universal semigroup compactification of G with respect to the property that the semigroup compactification contains a right zero element. We show that the topological center of this universal compactification is G. As an application of the previous results, we show that every closed ideal of GLUC contained in the ideal U(G) of uniform points of GLUC can be decomposed into 22κ(G) closed left ideals of GLUC.

A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains

Let $\Omega\subset R^N$ be a bounded open set with Lipschitz continuous boundary $\partial \Omega$. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on $L^2(\partial \Omega)$ which can also be ultracontractive. We also use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of a realization in $L^2(\Omega)$ of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.