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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]

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.

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.

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

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]

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]

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 \(\beta_t(x)\).