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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...

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.

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 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

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 $A$.

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.

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.

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.

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

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

We investigate which symmetric topological spaces have precisely one compatible nearness structure. We show how this situation exists for T1 topological spaces and for T2 topological spaces. The compatible nearness structures on a symmetric topological space form a sort of “interval” of structures with the topology itself as the topmost.

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fr\'echet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon., Comment: revised version

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d ${\cal N} = 4$ superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d ${\cal N}=4$ theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d ${\cal N} = 2$ subalgebra of the ${\cal N}=4$ algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on $S^3$ that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars., Comment: 108 pages; v2: typos corrected, some statements clarified

We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives., Comment: We added an outline of the proof of Corollary 4.9, and corrected some misprints

We prove that among the quasi-simple Lie groups only the group SL4(ℝ) occurs as the multiplication group of 3-dimensional connected topological loops L. These loops L are homeomorphic to the sphere S3. Moreover, there does not exist any connected topological loop having an at most 8-dimensional quasi-simple Lie groups as its multiplication group.

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.

Motivated by the concept of lattice-valued topological system of J.T. Denniston, A. Melton, and S.E. Rodabaugh, which extends lattice-valued topological spaces, this paper introduces the notion of lattice-valued bornological system as a generalization of lattice-valued bornological spaces of M. Abel and A. Sostak. We aim at (and make the first steps towards) the theory, which will provide a common setting for both lattice-valued point-set and point-free bornology. In particular, we show the algebraic structure of the latter.

[EN] In the paper one defines topological invariants of type degree for morphisms in the category Top₍₂₎ of topological pairs of spaces and continuous single valued maps, which admit homological n-spheres as target and arbitrary topological pairs of spaces as source. The different described degrees are acquired by means homological methods, and are a powerful tool in the root theory. Several existence theorems are obtained for equations with multivalued transformations., Project supported by CNEPRU No B00920110092 and PNR / ATRST No 8/u250/713. Laboratory M.M.E.R.E. U.M.1 Constantine and Russian FBR grant 11-01 -00382-a.

Let G be an admissible compact Hausdorff right topological group, that is, a group with a Hausdorff topology such that for each a ∈ G , the map g ↦ g a is continuous, and the set of a ∈ G such that the map g ↦ a g is continuous is dense in G. Such groups arise in the study of distal flows. In this paper we study the Fourier–Stieltjes algebra B ( G ) , the linear span of the continuous positive definite functions on G. We show that B ( G ) is isomorphic with the Fourier–Stieltjes algebra of an associated compact topological group. This result is then applied to obtain some geometric properties including the weak and weak ⁎ -fixed point properties on B ( G ) . We also study some related properties on the measure algebra M ( G ) .

In the present paper, we consider the problem of existence of nonequivalent definitions of topological transitivity, which is a classical problem of the topological dynamics. In particular, we use the fact that all available definitions of this kind imply a condition imposed on the dynamical system. The main result of our investigations is the complete classification of these dynamical systems.

“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

In this paper, we define new classes of sets called preopen sets, θI-semi-open sets, θI-β-open sets and θI-α-open sets in ideal topological spaces. Also, by using these sets, we obtain new decompositions of continuity in ideal topological spaces.

There exists a large extent of literature on the varieties of algebraic objects suth as groups and associative algebras. Earlier much attention has been focused around the varieties of topologico-algebraic objects such as topological groups, topological vector spaces and topological algebras. In this paper we consider the varieties of Banach algebras defined by polynomial identities and we show that There exists a one-to-one correspondence between the class of this varieties and closed T-ideals of the countably generated free algebra �� . Also we derive a result about the maximal ideals of �� .

Some equivalent conditions for a topological algebra to be a Q-algebra have been studied by several researchers. They have studied Q-algebras, mainly for unital topological algebras. In this paper some equivalent conditions are studied to be a Q-algebra for non-unital locally pseudoconvex algebras, locally A-pseudoconvex algebras and locally m-pseudoconvex algebras.

For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.

We prove that the liftings of a normal functor F in the category of compact Hausdorff spaces to the categories of (abelian) compact semigroups (monoids) are determined by natural transformations F(−)×F(−) → F(−×−) satisfying requirements that correspond to associativity, commutativity, and the existence of a unity. In particular, the tensor products for normal monads satisfy (not necessarily all) these requirements. It is proved that the power functor in the category of compacta is the only normal functor that admits a natural lifting to the category of convex compacta and their continuous affine mappings.

Motivated by the rapidly developing theory of lattice-valued (or, more generally, variety-based) topological systems, which takes its origin in the crisp concept of S. Vickers (introduced as a common framework for both topological spaces and their underlying algebraic structures--frames or locales), the paper initiates a deeper study of one of its incorporated mathematical machineries, i.e., the realm of soft sets of D. Molodtsov. More precisely, we start the theory of lattice-valued soft universal algebra, which is based in soft sets and lattice-valued algebras of A. Di Nola and G. Gerla. In particular, we provide a procedure for obtaining soft versions of algebraic structures and their homomorphisms, as well as basic tools for their investigation. The proposed machinery underlies many concepts of (lattice-valued) soft algebra, which are currently available in the literature, thereby enabling the respective researchers to avoid its reinvention in future.

We introduce the different notions of a new class of continuous functions called generalized semi Lambda (gs) continuous function in topological spaces. Its properties and characterization are also discussed.

Rational functions play an important role in linear system theory, and in the noncommutative case we mention in particular the works [225, 264, 265]. We also mention [65], which is set in the framework of noncommutative probability. There the approach uses a topological algebra of a very special structure (dual of a countably normed nuclear space, with a series of inequalities on the norms), and is also quite far from the quaternionic setting.

We introduce the notion of an essential copy in a topological space. Then we present a classification of topological spaces based on this notion. In addition, we obtain some results regarding this classification.