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

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.

The Topology and its Applications Research Group from the Instituto Universitario de Matematica Pura y Aplicada (IUMPA), Universitat Politecnica de Valencia, organizes the Workshop on Applied Topological Structures - WATS 2015. The main purpose of this workshop is to join experts in the application of several fields of general topology, including generalized metric and uniform structures, to other areas, with special emphasis in fixed point theory, topological algebra, functional analysis, fuzzy mathematics, computer science, etc

The Topology and its Applications Research Group from the Instituto Universitario de Matematica Pura y Aplicada (IUMPA), Universitat Politecnica de Valencia, organizes the Workshop on Applied Topological Structures - WATS 2015. The main purpose of this workshop is to join experts in the application of several fields of general topology, including generalized metric and uniform structures, to other areas, with special emphasis in fixed point theory, topological algebra, functional analysis, fuzzy mathematics, computer science, etc

The Topology and its Applications Research Group from the Instituto Universitario de Matematica Pura y Aplicada (IUMPA), Universitat Politecnica de Valencia, organizes the Workshop on Applied Topological Structures - WATS 2015. The main purpose of this workshop is to join experts in the application of several fields of general topology, including generalized metric and uniform structures, to other areas, with special emphasis in fixed point theory, topological algebra, functional analysis, fuzzy mathematics, computer science, etc

An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is relatively compact in the strong operator topology. A bounded linear operator on a Hilbert space is said to be strongly compact if the algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. First of all, some basic properties of strongly compact algebras are established. Next, a characterization of strongly compact normal operators is provided in terms of their spectral representation, and some applications are given. Finally, necessary and sufficient conditions for a weighted shift to be strongly compact are obtained in terms of the sliding products of its weights, and further applications are derived., Un álgebra de operadores lineales en un espacio de Hilbert se dice que es fuertemente compacta si su bola unidad es relativamente compacta en la topología fuerte de operadores. Un operador lineal y continuo en un espacio de Hilbert es fuertemente compacto si el algebra generada por el operador y la identidad es fuertemente compacta. Esta noción fue introducida por Lomonosov para estudiar el problema del subespacio invariante para operadores esencialmente normales. En primer lugar, se establecen algunas propiedades básicas de las álgebras fuertemente compactas. Se proporciona después una caracterizacion de los operadores normales fuertemente compactos en términos de su representación espectral y se dan algunas aplicaciones. Finalmente, se obtienen condiciones necesarias y suficientes para que un desplazamiento ponderado sea fuertemente compacto en términos de los productos deslizados de sus pesos. Se proporcionan algunas otras aplicaciones.

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. (C) 2005 Published by Elsevier B.V.

An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is relatively compact in the strong operator topology. A bounded linear operator on a Hilbert space is said to be strongly compact if the algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. First of all, some basic properties of strongly compact algebras are established. Next, a characterization of strongly compact normal operators is provided in terms of their spectral representation, and some applications are given. Finally, necessary and sufficient conditions for a weighted shift to be strongly compact are obtained in terms of the sliding products of its weights, and further applications are derived., Un álgebra de operadores lineales en un espacio de Hilbert se dice que es fuertemente compacta si su bola unidad es relativamente compacta en la topología fuerte de operadores. Un operador lineal y continuo en un espacio de Hilbert es fuertemente compacto si el algebra generada por el operador y la identidad es fuertemente compacta. Esta noción fue introducida por Lomonosov para estudiar el problema del subespacio invariante para operadores esencialmente normales. En primer lugar, se establecen algunas propiedades básicas de las álgebras fuertemente compactas. Se proporciona después una caracterizacion de los operadores normales fuertemente compactos en términos de su representación espectral y se dan algunas aplicaciones. Finalmente, se obtienen condiciones necesarias y suficientes para que un desplazamiento ponderado sea fuertemente compacto en términos de los productos deslizados de sus pesos. Se proporcionan algunas otras aplicaciones.

Given an algebraic theory T whose category of models is semi-abelian, we study the category Top T of topological models of T and generalize to it most classical results on topological groups. In particular, Top(T) 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. (C) 2004 Elsevier Inc. All rights reserved.

We present subsingular vectors of the N = 2 superconformal algebras other than the ones which become singular in chiral Verma modules, reported recently by Gato-Rivera and Rosado. We show that two large classes of singular vectors of the topological algebra become subsingular vectors of the antiperiodic NS algebra under the topological untwistings. These classes consist of BRST-invariant singular vectors with relative charges q = -2, - 1 and zero conformal weight, and no-label singular vectors with q = 0, - 1. In turn the resulting NS subsingular vectors are transformed by the spectral flows into subsingular and singular vectors of the periodic R algebra. We write down these singular and subsingular vectors starting from the topological singular vectors at levels 1 and 2. © 1999 Published by Elsevier Science B.V. All rights reserved.

Let M be a semifinite von Neumann algebra in a Hilbert space H and τ be a normal faithful semifinite trace on M. Let Mpr denote the set of all projections in M, e denote the unit of M, and ∥ · ∥ denote the C*-norm on M. The set of all τ-measurable operators M̃ with sum and product defined as the respective closures of the usual sum and product, is a *-algebra. The sets U(ε, δ)={x ε M̃: ∥xpk∥ ≤ ε and τ (e - p) ≤ δ for some p ε Mpr} ε>0; δ>0; form a base at 0 for a metrizable vector topology tτ on M̃, called the measure topology. Equipped with this topology, M̃ is a complete topological *-algebra. We will write xi τ→ x in case a net {xi} iεI ⊂ M̃ converges to x ε M̃ for the measure topology on M̃. By definition, a net {xi}iεI ⊂ M̃ converges τ-locally to x ε M̃ (notation: x i τl→ x) if xip τ→ xp for all p ε Mpr, τ(p) < ∞; and a net {xi} iεI ⊂ M̃ converges weak τ-locally to x ε M̃ (notation: xi wτl→ x) if xip τ→ pxp for all p ε Mpr, τ(p) < ∞.

Let M be a semifinite von Neumann algebra in a Hilbert space H and τ be a normal faithful semifinite trace on M. Let Mpr denote the set of all projections in M, e denote the unit of M, and ∥ · ∥ denote the C*-norm on M. The set of all τ-measurable operators M̃ with sum and product defined as the respective closures of the usual sum and product, is a *-algebra. The sets U(ε, δ)={x ε M̃: ∥xpk∥ ≤ ε and τ (e - p) ≤ δ for some p ε Mpr} ε>0; δ>0; form a base at 0 for a metrizable vector topology tτ on M̃, called the measure topology. Equipped with this topology, M̃ is a complete topological *-algebra. We will write xi τ→ x in case a net {xi} iεI ⊂ M̃ converges to x ε M̃ for the measure topology on M̃. By definition, a net {xi}iεI ⊂ M̃ converges τ-locally to x ε M̃ (notation: x i τl→ x) if xip τ→ xp for all p ε Mpr, τ(p) < ∞; and a net {xi} iεI ⊂ M̃ converges weak τ-locally to x ε M̃ (notation: xi wτl→ x) if xip τ→ pxp for all p ε Mpr, τ(p) < ∞.