2,183 results on '"Topological algebra"'
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52. Drinfeld and Jimbo’s Quantum Enveloping Algebras
- Author
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Kassel, Christian and Kassel, Christian
- Published
- 1995
- Full Text
- View/download PDF
53. References
- Author
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Navarro González, Juan A., Sancho de Salas, Juan B., Navarro González, Juan A., and Sancho de Salas, Juan B.
- Published
- 2003
- Full Text
- View/download PDF
54. New Perspectives in Algebra, Topology and Categories
- Abstract
This book provides an introduction to some key subjects in algebra and topology. It consists of comprehensive texts of some hours courses on the preliminaries for several advanced theories in (categorical) algebra and topology. Often, this kind of presentations is not so easy to find in the literature, where one begins articles by assuming a lot of knowledge in the field. This volume can both help young researchers to quickly get into the subject by offering a kind of « roadmap » and also help master students to be aware of the basics of other research directions in these fields before deciding to specialize in one of them. Furthermore, it can be used by established researchers who need a particular result for their own research and do not want to go through several research papers in order to understand a single proof. Although the chapters can be read as « self-contained » chapters, the authors have tried to coordinate the texts in order to make them complementary. The seven chapters of this volume correspond to the seven courses taught in two Summer Schools that took place in Louvain-la-Neuve in the frame of the project Fonds d’Appui à l’Internationalisation of the Université catholique de Louvain to strengthen the collaborations with the universities of Coimbra, Padova and Poitiers, within the Coimbra Group.
- Published
- 2021
55. ABOUT LOCALLY m-CONVEX ALGEBRAS WITH DENSE FINITELY GENERATED IDEALS.
- Author
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PEIMBERT, HUGO ARIZMENDI and PÉREZ-TISCAREÑO, REYNA MARÍA
- Subjects
- *
IDEALS (Algebra) , *COMMUTATIVE algebra , *CONVEX sets - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
56. On categories of merotopic, nearness, and filter algebras.
- Author
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Gompa, Vijaya L.
- Subjects
- *
ABELIAN groups , *NEARNESS spaces , *TOPOLOGICAL algebras - Abstract
We study algebraic properties of categories of Merotopic, Nearness, and Filter Algebras. We show that the category of filter torsion free abelian groups is an epireective subcategory of the category of filter abelian groups. The forgetful functor from the category of filter rings to filter monoids is essentially algebraic and the forgetful functor from the category of filter groups to the category of filters has a left adjoint. [ABSTRACT FROM AUTHOR]
- Published
- 2016
57. Subcategories of topological algebras.
- Author
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Gompa, Vijaya L.
- Subjects
- *
TOPOLOGICAL algebras , *FUNCTIONAL analysis - Abstract
In addition to exploring constructions and properties of limits and colimits in categories of topological algebras, we study special subcategories of topological algebras and their properties. In particular, under certain conditions, reective subcategories when paired with topological structures give rise to reective subcategories and epireective subcategories give rise to epireective subcategories. [ABSTRACT FROM AUTHOR]
- Published
- 2016
58. Topological Algebra via Inner Product.
- Author
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Azram, M.
- Subjects
- *
PROBABILITY measures , *FRECHET spaces , *BANACH algebras - Abstract
This paper is devoted to establish a probability measure on a unital commutative separable Frechet Q lmc*- algebra. Consequently a new technique to define an inner product on a unital commutative semi simple separable Frechet Q lmc*-algebra. We have shown that the resulting inner product space is a topological algebra. At the end we have established some properties of the introduced inner product. [ABSTRACT FROM AUTHOR]
- Published
- 2016
59. Generalized Convolution Behaviors and Topological Algebra.
- Author
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Bourlès, Henri and Oberst, Ulrich
- Subjects
- *
MATHEMATICAL convolutions , *TOPOLOGICAL algebras , *INTEGRAL domains , *MODULES (Algebra) , *COGENERATORS , *APPROXIMATION theory - Abstract
We investigate one-dimensional 'generalized convolution behaviors' (gen. beh.) that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Brethé, van Eijndhoven, Fliess, Gluesing-Luerssen, Habets, Loiseau, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space for these behaviors is the space E of smooth complex-valued functions on the real line. The ring of operators is the commutative integral domain E′ of distributions with compact support with its convolution product that acts on E by a variant of the convolution product and makes it an E′-module. Both E and E′ carry their standard topologies. Closed E′-submodules of finite powers of E were introduced and studied by Schwartz already in 1947 under the name 'invariant varieties' and are called gen. beh. here. A gen. beh. is called a behavior if it can be described by finitely many convolution equations. The ring E′ is not noetherian and therefore the standard algebraic arguments from one-dimensional differential systems theory have to be completed by methods of topological algebra. Standard constructions like elimination or taking (closed) images of behaviors may lead to gen. beh. and therefore the consideration of the latter is mandatory. It is not known whether all gen. beh. are indeed behaviors, but we show that many of them are, in particular all autonomous ones. The E′-module E is neither injective nor a cogenerator and, in particular, does not admit elimination in Willems' sense. But the signal submodule PE of all polynomial-exponential signals is injective for finitely generated modules and thus admits elimination. This is a useful replacement and approximation of the injectivity of E since the polynomial-exponential part of any gen. beh. is dense in it. We also describe a useful replacement of the cogenerator property and thus establish a strong relation between convolution equations and their solution spaces. Input/output structures of gen. beh. exist and are used to prove that also many nonautonomous generalized behaviors are indeed behaviors. The E′-torsion elements of E, i.e., the smooth functions which satisfy at least one nonzero convolution equation, are called 'mean-periodic functions' and were studied by many outstanding analysts. Their results are significant for gen. beh. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
60. On Exact Frames in Topological Algebras.
- Author
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Garg, Saakshi and Vashisht, L. K.
- Subjects
FUNCTIONAL analysis ,TOPOLOGY ,TOPOLOGICAL algebras - Abstract
We present necessary and sufficient conditions for a frame in topological algebras to be exact. [ABSTRACT FROM AUTHOR]
- Published
- 2016
61. Normalizers and Split Extensions.
- Author
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Bourn, Dominique and Gray, James
- Abstract
We make explicit a larger structural phenomenon hidden behind the existence of normalizers in terms of existence of certain precartesian maps related to the kernel functor. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
62. DERIVATIONS OF FRÉCHET NUCLEAR GB$^{\ast }$-ALGEBRAS.
- Author
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WEIGT, M. and ZARAKAS, I.
- Subjects
- *
TOPOLOGICAL algebras , *GELFAND-Naimark theorem , *QUANTUM theory , *LINEAR operators , *VECTOR spaces - Abstract
It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
63. On the Rational Homogeneous Manifold Structure of the Similarity Orbits of Jordan Elements in Operator Algebras
- Author
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Lorentz, Kai, Gohberg, I., editor, Bart, H., editor, and Kaashoek, M. A., editor
- Published
- 1991
- Full Text
- View/download PDF
64. Uniqueness of F-Algebra Topology for Commutative Semisimple Algebras
- Author
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Gurusamy Siva and Chinnadurai Ganesa Moorthy
- Subjects
F-algebra ,Topological algebra ,Bounded set ,010102 general mathematics ,Topology ,01 natural sciences ,010101 applied mathematics ,Metrization theorem ,Pharmacology (medical) ,Uniqueness ,0101 mathematics ,Fréchet algebra ,Commutative property ,Topology (chemistry) ,Mathematics - Abstract
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.
- Published
- 2019
65. Quantum integral equations of Volterra type in terms of discrete-time normal martingale
- Author
-
Yuling Tang and Jinshu Chen
- Subjects
symbols.namesake ,Topological algebra ,Discrete time and continuous time ,General Mathematics ,symbols ,Applied mathematics ,Uniqueness ,Discrete-time normal martingale,Volterra integral equation,operator,existence and uniqueness ,Martingale (probability theory) ,Quantum ,Integral equation ,Volterra integral equation ,Mathematics - Abstract
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.
- Published
- 2019
66. Orbifold construction for topological field theories
- Author
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Lukas Woike and Christoph Schweigert
- Subjects
Topological manifold ,Algebra and Number Theory ,Topological quantum field theory ,Topological algebra ,010102 general mathematics ,FOS: Physical sciences ,Cobordism ,Mathematical Physics (math-ph) ,Topology ,Mathematics::Algebraic Topology ,01 natural sciences ,Topological entropy in physics ,Homeomorphism ,Mathematics::K-Theory and Homology ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Topological ring ,Equivariant map ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics - Abstract
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
- Published
- 2019
67. On a problem of Bertram Yood
- Author
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Abel Mart and Abel Mati
- Subjects
topological ring ,advertive topological ring ,invertive topological ring ,simplicial topological ring ,yood problem ,topological algebra ,topological radical of a topological algebra ,Mathematics ,QA1-939 - Abstract
In 1964, Bertram Yood posed the following problem: whether the intersection of all closed maximal regular left ideals of a topological ring coincides with the intersection of all closed maximal regular right ideals of this ring. It is proved that these two intersections coincide for advertive and simplicial topological rings and, using this result, it is shown that the topological left radical and the topological right radical for every advertive and simplicial topological algebra coincide.
- Published
- 2014
- Full Text
- View/download PDF
68. On domains of unbounded derivations of generalized B $^{*}$ -algebras
- Author
-
Martin Weigt and Ioannis Zarakas
- Subjects
46K05 ,Pure mathematics ,46L05 ,Algebra and Number Theory ,46H05 ,Spectrum (functional analysis) ,derivation ,topological algebra ,Functional calculus ,$\mathrm{GB}^{*}$-algebra ,Locally convex topological vector space ,Domain (ring theory) ,46H35 ,Identity element ,Invariant (mathematics) ,Unit (ring theory) ,Analysis ,Mathematics ,Analytic function - Abstract
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.
- Published
- 2018
69. 'Analytical' functions of polynomial type
- Author
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Vladimir Todorov Todorov
- Subjects
Combinatorics ,Topological algebra ,Cover (topology) ,Partition of unity ,Operator algebra ,Lie algebra ,Field (mathematics) ,Differential operator ,Mathematics ,Separable space - Abstract
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.
- Published
- 2021
70. Effective topological complexity of spaces with symmetries
- Author
-
Marek Kaluba and Zbigniew Błaszczyk
- Subjects
Pure mathematics ,motion planning problem ,Topological algebra ,General Mathematics ,equivariant topological complexity ,01 natural sciences ,Mathematics::Algebraic Topology ,Equivariant topological complexity ,55M30 ,Motion planning problem ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Discrete mathematics ,Topological complexity ,68T40 ,Topological tensor product ,010102 general mathematics ,010101 applied mathematics ,Bounded function ,Homogeneous space ,55M30, 68T40 - Abstract
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
- Published
- 2021
71. Applied General Topology
- Subjects
topological dynamics ,topological algebra ,hyperspaces ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Published
- 2013
72. Certain Properties of $$n$$ -Characters and $$n$$ -Homomorphisms on Topological Algebras.
- Author
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Shayanpour, H., Honary, T., and Hashemi, M.
- Subjects
- *
HOMOMORPHISMS , *TOPOLOGICAL algebras , *BANACH algebras , *CAUCHY integrals , *STOCHASTIC convergence - Abstract
We extend the notion of homomorphisms and characters to $$n$$ -homomorphisms and $$n$$ -characters on algebras, and then show that some properties of characters are also valid for $$n$$ -characters on commutative $$lmc$$ topological algebras, and the space of continuous $$n$$ -characters $$M_{(A,n)}$$ is relatively compact in $$A'$$ (the dual space of $$A$$ ), with the weak* topology (Gelfand topology), whenever $$A$$ is a commutative $$lmc$$ $$Q$$ -algebra. We also find relations between characters, $$n$$ -characters, and continuous $$n$$ -characters on commutative Fréchet algebras. Let $$B$$ be a topological algebra and $$(A_{\alpha },\varphi _{\beta \alpha })$$ (resp. $$(A_{\alpha },\varphi _{\alpha \beta })$$ ) be an inductive system (resp. a projective system) of topological algebras. Then we obtain relations between $$n-Hom(A_{\alpha },B)$$ and $$n-Hom(A,B)$$ , or between [InlineEquation not available: see fulltext.], the inductive limit, and $$M_{(A,n)}$$ , where [InlineEquation not available: see fulltext.], is the inductive limit (resp. $$A=\varprojlim A_{\alpha }$$ , is the projective limit) and $$n-Hom(A_{\alpha },B)$$ (resp. $$n-Hom(A,B)$$ ), is the space of all continuous n-homomorphisms from $$A_\alpha $$ (resp. $$A$$ ) into $$B$$ . [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
73. C( M) as a smooth envelope of its subalgebras.
- Author
-
Akbarov, S.
- Subjects
- *
TOPOLOGICAL algebras , *FUNCTIONAL analysis , *BANACH algebras , *OPERATOR algebras , *APPROXIMATION theory , *FUNCTIONALS , *OPERATOR theory - Abstract
A smooth envelope of a topological algebra is introduced, and the following result is announced: the smooth envelope of a given subalgebra A in C( M) coincides with C( M) if and only if A has the same tangent bundle as M. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
74. HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS.
- Author
-
MAGHSOUDI, S. and SEOANE-SEPÚLVEDA, J. B.
- Subjects
- *
ALGEBRA , *TOPOLOGICAL algebras , *FUNCTIONAL analysis , *HYPERGROUPS , *GROUP theory - Abstract
Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
75. Banach Algebras and Applications
- Author
-
Tomasz Ciaś
- Subjects
Combinatorics ,Mathematics::Functional Analysis ,Series (mathematics) ,Topological algebra ,Domain (ring theory) ,Mathematics::General Topology ,Type (model theory) ,Mathematics::Representation Theory ,Compact operator ,Space (mathematics) ,Commutative property ,Noncommutative geometry ,Mathematics - Abstract
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 $\mathbb{R}^n$.
- Published
- 2020
76. Another note on effective descent morphisms of topological spaces and relational algebras
- Author
-
Maria Manuel Clementino and George Janelidze
- Subjects
Algebra ,Morphism ,Topological algebra ,Mathematics::Category Theory ,Topological tensor product ,Hausdorff space ,Geometry and Topology ,Topological space ,Relational algebra ,Monad ,Effective descent morphism ,Ultrafilter monad ,Locally finite space ,Alexandrov space ,Topological vector space ,Descent (mathematics) ,Mathematics - Abstract
We formulate two open problems related to and, in a sense, suggested by the Reiterman–Tholen characterization of effective descent morphisms of topological spaces.
- Published
- 2020
77. Locally Convex Quasi *-Algebras and their Representations
- Author
-
Fragoulopoulou M., Trapani C., Fragoulopoulou M., and Trapani C.
- Subjects
Settore MAT/05 - Analisi Matematica ,locally convex ,quasi *-algebra ,operator algebras ,topological algebra - Abstract
This book is a review of the work the authors have done in the past 20 years on the theory of locally convex quasi *-algebras
- Published
- 2020
78. ω-dominated function spaces and ω-bases in free objects of topological algebra
- Author
-
Taras Banakh and Arkady Leiderman
- Subjects
010101 applied mathematics ,Pure mathematics ,Topological algebra ,Function space ,010102 general mathematics ,Geometry and Topology ,0101 mathematics ,01 natural sciences ,Mathematics - Published
- 2018
79. Some aspects of dimension theory for topological groups
- Author
-
Alexander Arhangel’skii and J. van Mill
- Subjects
Topological manifold ,Topological algebra ,General Mathematics ,Covering group ,010102 general mathematics ,Locally compact group ,Homeomorphism group ,01 natural sciences ,Homeomorphism ,010101 applied mathematics ,Combinatorics ,Locally compact space ,0101 mathematics ,Mathematics ,Zero-dimensional space - Abstract
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.
- Published
- 2018
80. On the infinite-dimensional moment problem
- Author
-
Konrad Schmüdgen
- Subjects
Pure mathematics ,Topological algebra ,General Mathematics ,nuclear space ,Nuclear space ,Space (mathematics) ,01 natural sciences ,Carleman condition ,FOS: Mathematics ,Cylinder ,46G12 ,28C20 ,0101 mathematics ,Commutative property ,Mathematics ,Symmetric algebra ,010102 general mathematics ,symmetric algebra ,cylinder measure ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Moment problem ,Moment (mathematics) ,44A60 ,44A60 (primary), 46G12, 28C20 (secondary) ,moment problem - Abstract
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 $A$.
- Published
- 2018
81. Ambrosetti-Prodi type result to a Neumann problem via a topological approach
- Author
-
Elisa Sovrano
- Subjects
Multiplicity results ,Discrete mathematics ,Topological algebra ,Applied Mathematics ,010102 general mathematics ,Shooting method ,Topology ,01 natural sciences ,Topological entropy in physics ,Ambrosetti-Prodi problems ,Neumann series ,Neumman boundary conditions ,010101 applied mathematics ,symbols.namesake ,Von Neumann's theorem ,Von Neumann algebra ,Neumann boundary condition ,symbols ,Discrete Mathematics and Combinatorics ,Topological ring ,0101 mathematics ,Abelian von Neumann algebra ,Analysis ,Mathematics - Abstract
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.
- Published
- 2018
82. On topological complexity of non-orientable surfaces
- Author
-
Alexander Dranishnikov
- Subjects
Discrete mathematics ,Topological complexity ,Topological algebra ,Genus (mathematics) ,010102 general mathematics ,0103 physical sciences ,Topological ring ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We show that the topological complexity of the nonorientable surfaces of genus ≥4 is four.
- Published
- 2017
83. Topological Order in Physics
- Author
-
Ravi Karki
- Subjects
Physics ,Theoretical physics ,Topological algebra ,Topological degeneracy ,String-net liquid ,Topological order ,Quantum entanglement ,Symmetry protected topological order ,Topological entropy in physics ,Topological quantum number - Abstract
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)
- Published
- 2017
84. The kR-property in free topological groups
- Author
-
Chuan Liu, Fucai Lin, and Shou Lin
- Subjects
Connected space ,Pure mathematics ,Dense set ,Topological algebra ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010101 applied mathematics ,H-space ,Uniform continuity ,Topological ring ,Locally compact space ,0101 mathematics ,Mathematics ,Zero-dimensional space - Abstract
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.
- Published
- 2017
85. Certain strict topologies on the space of regular Borel measures on locally compact groups.
- Author
-
Maghsoudi, Saeid
- Subjects
- *
TOPOLOGY , *BOREL sets , *MEASURE theory , *COMPACT groups , *MATHEMATICAL complexes , *MATHEMATICAL convolutions - Abstract
Abstract: Let G denote a locally compact Hausdorff group and be the space of all bounded complex-valued regular Borel measures on G. In this paper, we define two strict topologies on and study various properties of these topologies such as metrizability, barrelledness and completeness. We also determine the dual space of and consider various continuity properties for the convolution product on under these topologies. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
86. Commutativity Results in Non Unital Real Topological Algebras.
- Author
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Oudadess, M. and Tsertos, Y.
- Subjects
- *
COMMUTATIVE algebra , *TOPOLOGICAL algebras , *MATHEMATICAL inequalities , *BANACH algebras , *ALGEBRA , *VECTOR topology - Abstract
We give conditions entailing commutativity in certain non unital real topological algebras. Several other results of complex algebras are also examined for real ones. [ABSTRACT FROM AUTHOR]
- Published
- 2012
87. Topological algebras with maximal regular ideals closed.
- Author
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Abel, Mati
- Abstract
It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
88. On the structure of graded Hilbert spaces
- Author
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Calderón Martín, Antonio J.
- Subjects
- *
HILBERT space , *ABELIAN groups , *UNITS of measurement , *ORTHOGONAL functions , *MATHEMATICAL analysis , *GROUP theory - Abstract
Abstract: Consider an arbitrary Hilbert space endowed with a continuous product which induces a grading on with respect to an abelian group G. We show that such a space has the form with U a closed subspace of (the factor associated to the unit element in G), and any a well described closed graded ideal of , satisfying if . Under certain conditions, the graded simplicity of is characterized and it is shown that is the closure of the orthogonal direct sum of the family of its minimal (closed) graded ideals, each one being a graded simple graded Hilbert space. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
89. Canonical Extensions and Discrete Dualities for Finitely Generated Varieties of Lattice-based Algebras.
- Author
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Davey, B. and Priestley, H.
- Abstract
The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show that topological and discrete dualities for $${\mathcal {A}}$$ exist in partnership. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
90. COMPACT MULTIPLICATION OPERATORS ON NONLOCALLY CONVEX WEIGHTED SPACES OF CONTINUOUS FUNCTIONS.
- Author
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MANHAS, J. S.
- Subjects
- *
HAUSDORFF measures , *MEASURE theory , *VECTOR spaces , *LINEAR algebra , *VECTOR analysis - Abstract
Let V be a system of weights on a completely regular Hausdorff space and let B(E) be the topological vector space of all continuous linear operators on a Hausdorff topological vector space E. Let CV0(X,E) and CVb(X,E) be the nonlocally convex weighted spaces of continuous functions. In the present paper, we characterize compact multiplication operators Mψ on CV0 (X,E) (or CVb(X,E)) induced by the operator-valued mappings ψ : X → B(E) (or the vector-valued mappings ψ : X → E, where E is a topological algebra). [ABSTRACT FROM AUTHOR]
- Published
- 2011
91. A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras
- Author
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Davey, B.A. and Priestley, H.A.
- Subjects
- *
TOPOLOGICAL algebras , *LATTICE theory , *ALGEBRAIC varieties , *FUNCTIONAL analysis , *ALGEBRAIC geometry , *MATHEMATICAL analysis - Abstract
Abstract: This paper investigates completions in the context of finitely generated lattice-based varieties of algebras. It is shown that, for such a variety , the order-theoretic conditions of density and compactness which characterise the canonical extension of (the lattice reduct of) any have truly topological interpretations. In addition, a particular realisation is presented of the canonical extension of A; this has the structure of a topological algebra whose underlying algebra belongs to . Furthermore, each of the operations of coincides with both the σ-extension and the π-extension of the corresponding operation on A, with which a canonical extension is customarily equipped. Thus, in particular, the variety is canonical, and all its operations are smooth. The methods employed rely solely on elementary order-theoretic and topological arguments, and by-pass the subtle theory of canonical extensions that has been developed for lattice-based algebras in general. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
92. Criteria of existence of bounded approximate identities in topological algebras.
- Author
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Podara, Christina P.
- Subjects
- *
EXISTENCE theorems , *APPROXIMATION theory , *BANACH algebras , *TOPOLOGICAL algebras , *COMMUTATIVE algebra , *FUNCTIONAL analysis - Abstract
Some results and criteria of existence concerning bounded approximate identities in Banach algebras are extended to the topological algebras setting. We thereby prove that the bidual of a commutative locally C*-algebra with either of the two Arens products is a unital commutative algebra, and that a quasinormable Fréchet m-convex algebra has a left (resp. right) bounded approximate identity if and only if it can be represented as an inverse limit of Banach algebras each of which has a left (resp. right) bounded approximate identity. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
93. REPRESENTATIONS OF TOPOLOGICAL ALGEBRAS BY PROJECTIVE LIMITS.
- Author
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ABEL, MATI
- Subjects
- *
TOPOLOGICAL algebras , *FUNCTIONAL analysis , *VECTOR topology , *MATHEMATICAL analysis , *VECTOR spaces , *BANACH algebras , *BANACH spaces , *COMPLEX variables , *GENERALIZED spaces , *GEOMETRY - Abstract
It is shown that a) it is possible to define the topology of any topological algebra by a collection of F-seminorms, b) every complete locally uniformly absorbent (complete locally A-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of metrizable locally uniformly absorbent algebras (respectively, A-(k-normed) algebras, where k ∈ (0, 1] varies, c) every complete locally idempotent (complete locally m-pseudoconvex) Hausdorff algebra is topologically isomorphic to a projective limit of locally idempotent Fréchet algebras (respectively, k-Banach algebras, where k ∈ (0, 1] varies) and every m-algebra is locally m-pseudoconvex. Condition for submultiplicativity of F-seminorm is given. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
94. Multipliers of Uniform Topological Algebras
- Author
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Mohammed El Azhari
- Subjects
Physics ,Mathematics::Functional Analysis ,Topological algebra ,Mathematics::Operator Algebras ,Multiplier algebra ,lcsh:Mathematics ,General Mathematics ,multiplier algebra ,uniform topological algebra ,General Medicine ,lcsh:QA1-939 ,Topology ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,locally m-pseudoconvex algebra ,FOS: Mathematics ,46H05, 46K05 - Abstract
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
- Published
- 2017
95. STUDY ON SOME TOPOLOGICAL GENERALIZED CLOSED GRAPHS
- Author
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D. Sheeba and N. Nagaveni
- Subjects
Combinatorics ,Connected space ,Topological algebra ,Topological graph theory ,Topological ring ,Closed graph theorem ,Geometry and Topology ,Topological entropy in physics ,Topological quantum number ,Homeomorphism ,Mathematics - Published
- 2017
96. Cross sections and pseudo-homomorphisms of topological abelian groups
- Author
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Hugo J. Bello, M.J. Chasco, and X. Domínguez
- Subjects
Topological algebra ,Covering group ,010102 general mathematics ,Topology ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,H-space ,Closed graph theorem ,Geometry and Topology ,Group homomorphism ,Topological group ,0101 mathematics ,Abelian group ,Quotient ,Mathematics - Abstract
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.
- Published
- 2017
97. Boundary-bulk relation in topological orders
- Author
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Xiao-Gang Wen, Hao Zheng, Liang Kong, Massachusetts Institute of Technology. Department of Physics, and Wen, Xiao-Gang
- Subjects
Physics ,Nuclear and High Energy Physics ,Topological algebra ,Strongly Correlated Electrons (cond-mat.str-el) ,Topological degeneracy ,010102 general mathematics ,Boundary (topology) ,FOS: Physical sciences ,Topology ,01 natural sciences ,Symmetry protected topological order ,Topological entropy in physics ,Condensed Matter - Strongly Correlated Electrons ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Topological order ,Quantum Algebra (math.QA) ,lcsh:QC770-798 ,lcsh:Nuclear and particle physics. Atomic energy. Radioactivity ,Uniqueness ,0101 mathematics ,010306 general physics ,Topological quantum number - Abstract
In this paper, we study the relation between an anomaly-free n+1 D topological order, which are often called n+1 D topological order in physics literature, and its n D gapped boundary phases. We argue that the n+1 D bulk anomaly-free topological order for a given n D gapped boundary phase is unique. This uniqueness defines the notion of the “ bulk ” for a given gapped boundary phase. In this paper, we show that the n+1 D “ bulk ” phase is given by the “center” of the n D boundary phase. In other words, the geometric notion of the “ bulk ” corresponds precisely to the algebraic notion of the “center”. We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the “ bulk ” satisfies the same universal property as that of the “center” of an algebra in mathematics, i.e. “ bulk = center”. The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions., National Science Foundation (U.S.) (Grant DMR-1506475), National Science Foundation (U.S.) (Grant NSFC11274192)
- Published
- 2017
98. Strict topology on locally compact groups
- Author
-
E. Fasahat and H. Samea
- Subjects
Topological algebra ,010102 general mathematics ,Subalgebra ,0102 computer and information sciences ,Group algebra ,Locally compact group ,Topology ,01 natural sciences ,010201 computation theory & mathematics ,Measure algebra ,Geometry and Topology ,Locally compact space ,Topological group ,0101 mathematics ,Mathematics ,Haar measure - Abstract
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.
- Published
- 2017
99. Topological protomodular algebras
- Author
-
Borceux, F. and Clementino, Maria Manuel
- Subjects
- *
TOPOLOGICAL groups , *CONTINUOUS groups , *TOPOLOGICAL algebras , *UNIVERSAL algebra - Abstract
Abstract: Topological groups have very striking properties, which have already been generalized to weaker “group like” structures, like various kinds of loops. This paper intends to show evidence that this generalization holds for a much wider class of theories, known as the protomodular theories, and which admit both an elegant categorical characterization and an easy description in universal algebra terms. Thus we propose a synthetic approach which allows to prove in a unique framework that the most striking properties of topological groups hold as well for loops or even semi-loops, rings with or without unit, associative algebras with or without unit, Lie algebras, Jordan algebras, Boolean algebras, Heyting algebras, Boolean rings, Heyting semi-lattices, and so on. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
100. On α Generalized Continuous Mappings in Ideal Topological Spaces
- Author
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S.Maragatha valli and D.Vino dhini
- Subjects
Pure mathematics ,Topological algebra ,Topological tensor product ,Locally convex topological vector space ,Compact-open topology ,Topological space ,Topological vector space ,Homeomorphism ,Mathematics ,Continuous linear operator - Published
- 2017
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