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We present conditions under which the Gel’fand transform E ∧ , of locally m-convex algebra E, is a dense subalgebra of C c (M (E)) . The partition of unity and the local theorem are given for a commutative unital locally m-convex algebra. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the weighted generalization of the space of continuous functions vanishing at infinity and correct some wrong results in the paper. Let X be a locally compact space and ν is an arbitrary weight (non‐negative function) on X. We give a correct and comprehensive definition of the weighted generalization C0ν(X)$C_0^\nu (X)$ of C0(X)$C_0(X)$, and show that it is a seminormed space with respect to the canonical seminorm ∥f∥ν=supx∈X|f(x)|$\Vert f\Vert _\nu =\sup _{x\in X}|f(x)|$, where f∈C0ν(X)$f\in C_0^\nu (X)$. We find conditions on ν under which C0ν(X)$C_0^\nu (X)$, with respect to ∥.∥ν$\Vert.\Vert _\nu$, becomes a normed space or a Banach space or an algebra, or a topological algebra, respectively. [ABSTRACT FROM AUTHOR]

This paper deals with the automatic continuity of multiplicative polynomial operators on a class of topological algebras. Several results are derived in this direction. We also support our results by some examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present a general survey on formal power series over topological algebras, along with some perspectives which are not easily found in the literature. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We concider so called block-symmetric polynomials on sequence spaces ℓ1 ⊕ ℓ∲, ℓ1 ⊕ c, ℓ1 ⊕ c0, that is, polynomialswhich are symmetricwith respect to permutations of elements of the sequences. It is proved that every continuous block-symmetric polynomials on ℓ1 ⊕ ℓ∲ can be uniquelly represented as an algebraic combination of some special block-symmetric polynomials, which form an algebraic basis. It is interesting to note that the algebra of block-symmetric polynomials is infinitegenerated while ℓ∲ admits no symmetric polynomials. Algebraic bases of the algebras of blocksymmetric polynomials on ℓ1 ⊕ ℓ∲ and ℓ1 ⊕ c0 are described. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras. [ABSTRACT FROM AUTHOR]

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 3D 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 multiquons and their transformations in a natural way. We obtain a 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, which are widely used in tensor networks. We obtain a 3D representation of the controlled NOT (CNOT) gate that is considerably simpler than earlier work, and a 3D topological protocol for teleportation. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Letbe open with. We characterize the setshaving the following property: for every real analytic functionfonwith Taylor expansionat zero, the seriesis also the Taylor expansion at zero of some real analytic function on. This result gives a characterization of the idempotents in the algebraof Hadamard-type operators on the space of all real analytic functions, i.e. operators with all monomials being eigenvectors. In many cases, we also describe the multiplicative functionals onand the (continuous) algebra homomorphisms. We show that the algebrais never locally m-convex and in many cases it is not a Q-algebra. [ABSTRACT FROM AUTHOR]

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]

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]

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]

We present necessary and sufficient conditions for a frame in topological algebras to be exact. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

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]

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]

The “Gel’fand sheaf” of a topological algebra is endowed with a uniform structure, this being complete if and only if, the spectrum of the algebra considered is complete. Examples are also provided. [ABSTRACT FROM AUTHOR]

We review our recent formulation of Colombeau-type algebras as Hausdorff sequence spaces with ultranorms, defined by sequences of exponential weights. We extend previous results and give new perspectives related to echelon-type spaces, possible generalizations, asymptotic algebras, concepts of association, and applications thereof. [ABSTRACT FROM AUTHOR]

Given a simplicial topologically non radical algebra A, we characterize its topological radical, radA. If furthermore A is advertive, then radA coincides with the Jacobson radical RadA. On the other hand, it is shown that every two-sided invertive simplicial topological Gelfand-Mazur algebra has a functional spectrum and for every topologically nonradical simplicial Gelfand-Mazur amits the set X (A), of all continuous multiplicative linear functionals, is not empty. [ABSTRACT FROM AUTHOR]

Discusses the gap between the abstract and concrete models of computation in the topological algebras. Introduction of the concept of approximable many-valued computation on metric algebras; Ways to define the computable functions on topological algebras; Details on the standard concrete model of Grzegorczyk-Lacombe computability.

We show that a locally convex algebra is topologically algebraic if, and only if, it is algebraic. [ABSTRACT FROM AUTHOR]

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ, μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ β and μ β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ, μ), τ) is always sequentially complete locally convex topological algebra. [ABSTRACT FROM AUTHOR]