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In this paper, we consider the notion of quasi-multipliers to the general topological algebra setting, not necessarily normed or locally convex. We discuss the quasi-uniform and quasi-strong operator topologies on the algebra QM(A) of all bilinear and jointly continuous quasi-multipliers on A and study their various properties.

We show that in a noetherian commutative unital topological algebra, the prime ideals associated with a closed ideal as well as its isolated primary components are closed. We obtain a version of the closed graph theorem. An example of a noetherian (even principal) commutative unital semi-simple and incomplete normed algebra whose each ideal is closed is also given.

If X is Hausdorff topological space and Cc (X) is the topological algebra obtained by endowing the algebra C(X) of all continuous functions on X with the topology τc of uniform convergence on the c...

Let \(A\) be a topological algebra and \(\beta\) a subadditive boundedness radius on \(A\). In this paper we show that \(\beta\) is, under certain conditions, automatically submultiplicative. Then we apply this fact to prove that the spectrum of any element of \(A\) is non-empty. Finally, in the case when \(A\) is a normed algebra, we compare the initial normed topology with the normed topology \(\tau_{\beta}\), induced by \(\beta\) on \(A\), where \(\beta^{-1} (0)=0\).

First, this paper introduces the notion of L0-convex compactness for a special class of closed convex subsets–closed L0-convex subsets of a Hausdorff topological module over the topological algebra...

Let $X$ be a weighted direct sum of infinity many copies of complex spaces $\ell_1\bigoplus \ell_1.$ We consider an algebra consisting of polynomials on $X$ which are supersymmetric on each term $\ell_1\bigoplus \ell_1.$ Point evaluation functionals on such algebra gives us a relation of equivalence `$\sim$' on $X.$ We investigate the quotient set $X/\sim$ and show that under some conditions, it has a real topological algebra structure.

The set $$\mathcal {E}'(\mathbb {R})$$ E ′ ( R ) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of $$\mathcal {E}'(\mathbb {R})$$ E ′ ( R ) is 2.

Let A be a complex commutative semisimple complete LMC algebra with respect to a topology $$\tau _{1}$$ and a complete metrizable topological algebra with respect to a topology $$\tau _{2}$$ . It is proved that every $$\tau _{1}$$ -bounded set is a $$\tau _{2}$$ -bounded set. This generalizes a result of R. L. Carpenter on uniqueness of Frechet algebra topology for complex commutative semisimple algebras.

In this paper, we aim to introduce a quantum linear stochastic Volterra integral equation of convolution type with operator-valued kernels in a nuclear topological algebra. We first establish the existence and uniqueness of the solutions and give the explicit expression of the solutions. Then we prove the continuity, continuous dependence on free terms and other properties of the solution.

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite groups assigns in a functorial way to a $G$-equivariant topological field theory an $H$-equivariant topological field theory, the pushforward theory. When $H$ is the trivial group, this yields an orbifold construction for $G$-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization., 21 pages, accepted for publication in the Journal of Pure and Applied Algebra

We study properties under which the domain of a closed derivation $\delta:D(\delta)\rightarrow A$ of a generalized B $^{*}$ -algebra $A$ remains invariant under analytic functional calculus. For a complete, generalized B $^{*}$ -algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of $A$ belongs to the domain of the derivation, along with a condition related to the coincidence $\sigma_{A}(x)=\sigma_{D(\delta)}(x)$ of the (Allan) spectra for every element $x\in D(\delta)$ . Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed $*$ -derivation $\delta$ of a complete GB $^{*}$ -algebra with jointly continuous multiplication such that $1\in D(\delta)$ and $x$ a normal element of the domain, $f(x)\in D(\delta)$ for every analytic function on a neighborhood of the spectrum of $x$ . We also give an example of a closed derivation of a GB $^{*}$ -algebra which does not contain the identity element. A condition for a closed derivation of a GB $^{*}$ -algebra $A$ to be the generator of a one-parameter group of automorphisms of $A$ is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.

We deal in this paper with functions of the type f(t,x)=∑k=0∞ak(t)xk where t ∈ T, x ∈ X. In what follows we shall consider different types of spaces T it supposed to be a separable metric space of different kinds and X, which suppose to be some kind of a topological algebra. Next suppose that ak(t)k∈ℕ+ is a partition of unity. More generally cα(t)α∈N+n we consider here a complex - valued (if necessary) partition of unity in ℝn with multi-indexes if n > 1. A partition of unity is locally finite hence the function f (t, x) is a polynomial concerning the variable x which should explain the name of this note.Note that x it may belong to various algebras, Banach one, operator algebras, functional algebras one etc. The aim of this note is to see how this point of view may help to solve some problems of mathematical physics in non-standard way. It may be Banach of different types, some Lie algebra etc. Here is our basic stock of examples:1) Consider a topological space T and countable partition of unity {ak (t)}, (k ∈ N+) in it. Clearly that there is a locally finite open (or even point finite) cover U = {Uk} (k ∈ N+) for which supp (ak) ⊂ Uk. Because ordt U < ∞ we have ak ≡ 0 for almost all k and thus f(t,x)=∑k=0∞ak(t)xk is a polynomial of x ∈ X for every point t ∈ T. Note that we suppose here that X admits algebraic structure a field or a module or a ring, and it is not necessary to be loaded with some topology.2) In addition, we can consider some structures on T or X. For example, if T = ℝn and ak(t) ∈ C∞(ℝn) then one may consider derivatives ∂tα f(t, x), so ∂αf(t,x)=∑k=0∞∂tαak(t)xk where α=(α1,α2,…,αn)∈ℕ+n is a multi-index. Moreover, if X is an n-dimensional Euclidean space ℝn one can consider the function f(t,x)=∑| α |=0∞aα(t)xα where as usual xα=x1α1x2α2⋯xnαn follows next then for an arbitrary differential operator D we haveDf(t,x)=∑| α |=0∞D(aα(t)xα);(1)note that we deal here with finite sums. In the following text we discuss some properties and examples of ”analytical” functions of polynomial type.

We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant, Dranishnikov and Lubawski-Marzantowicz. In particular, it is bounded from above by Farber's topological complexity., Comment: New title; a short section with open problems included at the end of the paper. Numerous minor improvements throughout the text. Final version, to appear in Publ. Mat. 19 pages, 2 figures

Let $\mathscr{L}^*(s)$ be the maximal $\mathcal{O}^*$-algebra of unbounded operators on $\ell_2$ whose domain is the space $s$ of rapidly decreasing sequences. This is a noncommutative topological algebra with involution which can be identified, for instance, with the algebra $\mathscr L(s)\cap\mathscr L(s')$ or the algebra of multipliers for the algebra $\mathscr{L}(s',s)$ of smooth compact operators. We give a simple characterization of unital commutative Fr\'echet ${}^*$-subalgebras of $\mathscr{L}^*(s)$ isomorphic as a Fr\'echet spaces to nuclear power series spaces $\Lambda_\infty(\alpha)$ of infinite type. It appears that many natural Fr\'echet ${}^*$-algebras are closed ${}^*$-subalgebras of $\mathscr{L}^*(s)$, for example, the algebras $C^\infty(M)$ of smooth functions on smooth compact manifolds and the algebra $\mathscr S (\mathbb{R}^n)$ of smooth rapidly decreasing functions on $\mathbb{R}^n$.

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of [ 0 , 1 ] . And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let G be a topological group with a countable network. Does it follow that dim G = ind G = Ind G ? The same question if X is a compact coset space. We also do not know whether the inequality dim ( G × H ) ≤ dim G + dim H holds for arbitrary topological groups G and H which are subgroups of σ -compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature.

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra $A$. We define moment functionals on $A$ as linear functionals which can be written as integrals over characters of $A$ with respect to cylinder measures. Our main results provide such integral representations for $A_{+}$–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application, we solve the moment problem for the symmetric algebra $S(V)$ of a real vector space $V$. As a byproduct, we obtain new approaches to the moment problem on $S(V)$ for a nuclear space $V$ and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra $A$.

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation \begin{document}$u''+f(x, u(x))=μ$\end{document} when the nonlinearity has the following form: \begin{document}$f(x, u):=a(x)g(u)-p(x)$\end{document} . The assumptions considered generalize the classical one, \begin{document}$f(x, u)\to+∞$\end{document} as \begin{document}$|u|\to+∞$\end{document} , without requiring any uniformity condition in \begin{document}$x$\end{document} . The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.

We show that the topological complexity of the nonorientable surfaces of genus ≥4 is four.

In general, we know that there are four states of matter solid, liquid, gas and plasma. But there are much more states of matter. For e. g. there are ferromagnetic states of matter as revealed by the phenomenon of magnetization and superfluid states defined by the phenomenon of zero viscosity. The various phases in our colorful world are so rich that it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. Topological phenomena define the topological order at macroscopic level. Topological order need new mathematical framework to describe it. More recently it is found that at microscopic level topological order is due to the long range quantum entanglement, just like the fermions fluid is due to the fermion-pair condensation. Long range quantum entanglement leads to many amazing emergent phenomena, such as fractional quantum numbers, non- Abelian statistics ad perfect conducting boundary channels. It can even provide a unified origin of light and electron i.e. gauge interactions and Fermi statistics. Light waves (gauge fields) are fluctuations of long range entanglement and electron (fermion) are defect of long range entanglements.The Himalayan Physics Vol. 6 & 7, April 2017 (108-111)

A space X is called a k R -space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is continuous. In this paper, we mainly discuss the k R -property in the free topological groups, and generalize some well-known results of K. Yamada.

Let $E$ be a complete uniform topological algebra with Arens-Michael normed factors $\left(E_{\alpha}\right)_{\alpha\in\Lambda}.$ Then $M\left(E\right) \cong \varprojlim M\left(E_{\alpha}\right)$ within an algebra isomorphism $\varphi$. If each factor $E_{\alpha}$ is complete, then every multiplier of $E$ is continuous and $\varphi$ is a topological algebra isomorphism where $M\left(E\right)$ is endowed with its seminorm topology., Comment: 8 pages

We say that a mapping ω between two topological abelian groups G and H is a pseudo-homomorphism if the associated map ( x , y ) ∈ G × G ↦ ω ( x + y ) − ω ( x ) − ω ( y ) ∈ H is continuous. This notion appears naturally in connection with cross sections (continuous right inverses for quotient mappings): given an algebraically splitting, closed subgroup H of a topological group X such that the projection π : X → X / H admits a cross section, one obtains a pseudo-homomorphism of X / H to H, and conversely. We show that H splits as a topological subgroup if and only if the corresponding pseudo-homomorphism can be decomposed as a sum of a homomorphism and a continuous mapping. We also prove that pseudo-homomorphisms between Polish groups satisfy the closed graph theorem. Several examples are given.

Let G be a locally compact group, A a subalgebra of the measure algebra M ( G ) , and A a family of Borel subsets of G that is closed under finite unions. In this paper, among other results, we find sufficient conditions on A , that imply A is a semi-topological algebra with respect to the strict topology β A . We also find necessary and sufficient conditions on G, that imply A is a topological algebra with respect to the strict topology β A , where A is a family of Borel subsets of G with finite Haar measure.

If X is a topological space and A X, then the number of distinct sets that can be obtained from A by using all possible compositions for operators i , c (where = ; ; ; ) introduced by Cs asz ar is at the most 25. Explicit expressions for these sets are provided. An example is provided where all the 25 di erent sets are determined. The result is also discussed for special cases such as when the space is extremally disconnected, resolvable, open-unresolvable, and partition spaces.

In this paper we study separation axioms and connected properties on (semi)topological quotient BCK-algebras. We bring some conditions which under a (semi)topological quotient BCK-algebra have at least one of the topological properties $$T_1,$$ Hausdorff, regular, normal, connected, locally connected, totally disconnected space.

Left Bol loops with the A l -property or gyrogroups are generalization of groups which do not explicitly have associativity. In this work, we define topological gyrogroups and study some properties of them. In spite of having a weaker algebraic form, topological gyrogroups carry almost the same basic properties owned by topological groups. In particular, we prove that being T 0 and T 3 are equivalent in topological gyrogroups. Furthermore, a topological gyrogroup is first countable if and only if it is premetrizable. Finally, a direct product of topological gyrogroups is a topological gyrogroup.

The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined using open covers. Furthermore, a quantitative analogue of Bowen's theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Biś. Finally, a formula for the entropy of skew-product transformation with respect to the subshift is obtained.

It is shown that for any ordinal α, there exists a T 0-space whose d-rank is equal to α.

In this paper, the notions of (para, quasi, semi) topological MV-algebras are defined and their related properties are studied. Also, topologies with which an MV-algebra can be a (para, semi) topological MV-algebra are obtained. Clearly, a topological MV-algebra is a (para, quasi, semi) topological MV-algebra, but the converse is not true, as shown by an example. In addition, we study ideals and filters in (para, quasi) topological MV-algebras, and we show that a quasitopological MV-algebra is a topological MV-algebra if the ideal {0}, or equivalently, the filter {1} is open.

International audience; Let K be an ultrametric complete algebraically closed field, let D be a disk {x ∈ K | |x| < R} (with R in the set of absolute values of K) and let A be the Banach algebra of bounded analytic functions in D. The vector space generated by the set of characters of A is dense in the topological dual of A if and only if K is not spherically complete. Let H(D) be the Banach algebra of analytic elements in D. The vector space generated by the set of characters of H(D) is never dense in the topological dual of H(D).

We obtain explicitly principal extensions of analytic functions of the complex variable into an infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We consider an extension of differentiable in the sense of Gâteaux functions with values in a topological vector space being an expansion of the mentioned algebra and its relations to spatial potentials.

We present a 3D, topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a three-dimensional manifold with boundary. A quon is a composite that acts as a particle. Specifically a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multi-quons and their transformations in a natural way. We obtain a new type of relation, a string-genus "joint relation," involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the $C^{*}$ Hopf algebra relations, that are widely used in tensor networks. We obtain a 3D representation of the Controlled NOT or CNOT gate (that is considerably simpler than earlier work) and a 3D topological protocol for teleportation.

One of the main obstacles before the development of the theory of topological hypergroups is the fact that translation of open sets may not be open in this setting. In this paper, we get rid of such obstacle by introducing the concept of topological complete hypergroups and investigate some of their properties.

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).

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the exis-tence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common xed point property. Mathematics Subject Classicatio n (2010): Primary 43A60, 46H05; Secondary 43A07, 46L10, 47L10. Key words : Lau algebra, left invariant mean, left introverted subspace, semi-topological semigroup, weakly almost periodic functional.

We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang-Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological invariants. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories of effective models are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems., This work will appear in a special Nuclear Physics issue "Memoriam Raymond Stora"

After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologi es on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities a nd provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.

This paper is a part of the results of my study on lattice topology on some algebraic structures; for examples on Rn (the collection of all n-tuples of real numbers), S (R) (the collection of all sequences of real numbers), C[a, b] (the collection of all continuous functions on [a, b]), and M[a, b] (the collection of all measurable functions on [a, b]). Each of them is a lattice topological algebra (See [6]) and each of them can be considered as a generalization of the system of real numbers.This paper is a part of the results of my study on lattice topology on some algebraic structures; for examples on Rn (the collection of all n-tuples of real numbers), S (R) (the collection of all sequences of real numbers), C[a, b] (the collection of all continuous functions on [a, b]), and M[a, b] (the collection of all measurable functions on [a, b]). Each of them is a lattice topological algebra (See [6]) and each of them can be considered as a generalization of the system of real numbers.

In Chap. 1 we recalled some of the basic notions and results which are commonly used in differential geometry. We had presented them with the intent of showing how they pass into non-commutative geometry. By definition, a non-commutative space is a spectral triple\(\{ \mathcal {A}, \rho , F \}\) consisting of an associative, not necessarily commutative or topological algebra \(\mathcal {A}\), a Fredholm operator F acting on a separable Hilbert space H and \(\rho : \mathcal {A} \longrightarrow \mathit {L}(H)\) a representation of the algebra \(\mathcal {A}\) onto the Hilbert space H, subject to additional conditions. Such a structure codifies an abstract elliptic operator defined by Atiyah (K-Theory, Benjamin, 1967). While in differential geometry elliptic operators are defined after multiple structures are summed up, in non-commutative geometry this process is reversed. The study of non-commutative geometry consists of finding the hidden mathematical structures codified by a spectral triple. In Chap. 2 we show how the notions of space, bundles, homology/cohomology and characteristic classes can be extracted from spectral triples. We stress that non-commutative geometry objects are defined in such a way that (1) the locality and (2) the commutativity assumptions, used in the differentiable geometry counterparts, are not postulated.

Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schr\"odinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra., Comment: 19 pages

ICFA mini-workshop on Nonlinear Dynamics and Collective Effects in Particle Beam Physics, NOCE2017, Arcidosso, Italy, 19 Sep 2017 - 22 Sep 2017; 12 pp. (2017)., Here I rephrase some of the results of work performed in several collaborations with K.Heinemann and J.A.Ellison (University of New Mexico, Department of Mathematics, Albuquerque, NM, USA), D.P.Barber (DESY; also visiting staff member at the University of Liverpool, Liverpool, UK, EU), and A.Kling (University of Applied Sciences Osnabrueck, Lingen, Germany, EU) on a generalized look on spin dynamics and beam polarization in storage rings. It is done in a way that emphasizes the applicability of the concepts to real world polarized beams rather than presenting the results in their most general form. The latter view can be found in several articles on the ArXiv and will be published in refereed journals soon.I will introduce several ``spin-related'' systems, state some selected main results of the above mentioned work and then recover and compare some basic (and some not so basic) findings for the various systems in the light of our generalized approach.

The property of having a finite number of orbits under the regular action has been used to study properties of rings and algebras. For example, in ring theory, Yasuyuki Hirano was able to use this property to show that rings with finitely many orbits under the regular action can be decomposed as direct sum of uniserial rings and a finite ring. In this thesis, we study modules under the regular action. More precisely, if R is a unital ring and M is a left(right) R-module, we describe all modules M that have finitely many orbits under the regular action. Along the way, we give a (new) module theoretical proof to the theorem of Yasuyuki Hirano on the classification of rings with finitely many orbits under the regular action which was proven using using methods from ring theory. Our charaterization of modules with finitely many orbits under the regular action shows a connection between algebras with finitely many submodules and distributive modules. A particular algebra that is of interest to us is the incidence algebra of a finite poset. Incidence algebras were originally introduced in the 1960’s by Gian-Carlo Rota as a way to study combinatorial problems but it became apparent later on that such algebras were an interesting object to study in their own right. They include ring theoretical examples such as the product of copies of a ring R and the upper triangular matrices over R. Robert B. Feinberg in his work on incidence algebras developed an internal characterization of incidence algebras of lower finite quasi-ordered sets. For example, he showed that an associative unital complete topological algebra Λ over a field k, where k has the discrete topology, is isomorphic to an incidence algebra if and only if 1. Λ has a faithful unital left module M with a distributive lattice of submodules. Fur

The notion of a weakly locally compact topological abelian group introduced in this paper generalizes the notion of a fibrous topological abelian group studied by N. Ya. Vilenkin.

Recently, J.~T.~Denniston, A.~Melton, and S.~E.~Rodabaugh introduced a lattice-valued analogue of the concept of institution of J.~A.~Goguen and R.~M.~Burstall, comparing it, moreover, with the (lattice-valued version of the) notion of topological system of S.~Vickers. In this paper, we show that a suitable generalization of topological systems provides a convenient framework for doing certain kinds of (lattice-valued) institutions.

It is well known, as a consequence of a theorem of Richard Arens, that a commutative Fréchet locally $m$ -convex algebra $E$ with unit does not have dense finitely generated ideals. We shall see that this result can no longer be true if $E$ is not complete and metrizable. We observe that the same is true for the theorem of Arens; that is, this theorem can no longer be true if $E$ is not complete and metrizable. Moreover, several conditions for a unital commutative (not necessarily complete) locally $m$ -convex algebra are given, for which all maximal ideals have codimension one.