Search

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman's theorem can be derived from it. [ABSTRACT FROM AUTHOR]

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]

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.

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]

This paper presents dual B-topologies that are determined by filterbase and some sets in a dual B-algebra. Also, some properties of a filterbase in a dual B-topological space are provided. In particular, a commutative dual B-topological space and a symmetric B-topological space are topological dual B-algebras. [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]

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 $\beta^{-1} (0)=0$.

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

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.

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. [ABSTRACT FROM AUTHOR]

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]

Properties of the sets of left, right, and two-sided topologically quasi-invertible elements, topological spectra, and topological spectral radii of elements in (not necessarily unital or commutative) topological algebras are studied. We prove the spectral mapping theorem for the topological spectrum of elements in commutative complex (not necessarily unital) topological algebras and show that the topological spectral radius (as a map) is a submultiplicative seminorm in a topological algebra with a functional topological spectrum. [ABSTRACT FROM AUTHOR]

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]

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. [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]

Main properties of the topology defined by a bornology on a topological linear space and main properties of Mackey Q-algebras are presented. Relationships of Mackey Q-algebras with other classes of topological algebras are described. It is shown that every Mackey Q-algebra is an advertibly Mackey complete algebra, every strongly sequential Mackey Q-algebra is a Q-algebra, every infrasequential Mackey Q-algebra is an advertibly complete algebra, and every infrasequential advertive Hausdorff algebra is a Mackey Q-algebra. [ABSTRACT FROM AUTHOR]

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 prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is equivalent to the underlying topological space being a Stone space. Condensing the core idea of those classical results, we introduce the concept of affine boundedness for an arbitrary universal algebra and show that for an affinely bounded topological algebra over a compact signature profiniteness is equivalent to the underlying topological space being a Stone space. Since groups, semigroups, rings, and distributive lattices are indeed affinely bounded algebras over finite signatures, all these known cases arise as special instances of our result. Furthermore, we present some additional applications concerning topological semirings and their modules, as well as distributive associative algebras. We also deduce that any affinely bounded simple compact algebra over a compact signature is either connected or finite. Towards proving the main result, we also establish that any topological algebra is profinite if and only if its underlying space is a Stone space and its translation monoid is equicontinuous. [ABSTRACT FROM AUTHOR]

In this paper we prove that the extended spectrum \(\Sigma(x)\), defined by W. Żelazko, of an element \(x\) of a pseudo-complete locally convex unital complex algebra \(A\) is a subset of the spectrum \(\sigma_A(x)\), defined by G.R. Allan. Furthermore, we prove that they coincide when \(\Sigma(x)\) is closed. We also establish some order relations between several topological radii of \(x\), among which are the topological spectral radius \(R_t(x)\) and the topological radius of boundedness \(\beta_t(x)\).

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]

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]

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]

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]

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.

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 an algebraic theory T whose category of models is semi-abelian, we study the category TopT of topological models of T and generalize to it most classical results on topological groups. In particular, TopT is homological, which includes Barr regularity and forces the Mal''cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, connected, totally disconnected and profinite T-algebras. [Copyright &y& Elsevier]

Data such as real and complex numbers, discrete and continuous time data streams, waveforms, scalar and vector fields, and many other functions, are fundamental for many kinds of computation. In the theory of data, such data types are modelled using topological, or metric, many-sorted algebras and continuous homomorphisms. A theory of such topological data types is needed to answer the general questions:Such a theory seems to be in its infancy: there are many approaches to computability theory on general and specific spaces, and few approaches to specification theory. In some earlier papers, we have studied the questions 1 and 2 with the needs of data type theory in mind, and built a bridge between computations and specifications to try to answer 3. In this paper, we extend and combine several of our results, to prove new theorems thatA technical issue is the localisation of uniform continuity using exhaustions of open sets. We use specifications composed of conditional equations, inequalities and, for convenience, new exhaustion primitives, that define functions uniquely up to isomorphism. [Copyright &y& Elsevier]