The Experts below are selected from a list of 129 Experts worldwide ranked by ideXlab platform
Taras Banakh - One of the best experts on this subject based on the ideXlab platform.
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omega omega dominated function spaces and omega omega bases in free objects of Topological Algebra
arXiv: General Topology, 2016Co-Authors: Taras Banakh, Arkady LeidermanAbstract:A Topological space $X$ is defined to have an $\omega^\omega$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_\alpha[x])_{\alpha\in\omega^\omega}$ such that $U_\beta[x]\subset U_\alpha[x]$ for all $\alpha\le\beta$ in $\omega^\omega$. We characterize Topological and uniform spaces whose free (locally convex) Topological vector spaces or free (Abelian or Boolean) Topological groups have $\omega^\omega$-bases.
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fans and their applications in general topology functional analysis and Topological Algebra
arXiv: General Topology, 2016Co-Authors: Taras BanakhAbstract:A family of closed subsets of a Topological space $X$ is called a (strict) $Cld$-fan in $X$ if this family is (strictly) compact-finite but not locally finite in $X$. Applications of (strict) $Cld$-fans are based on a simple observation that $k$-spaces contain no $Cld$-fan and Ascoli spaces contain no strict $Cld$-fan. In this paper we develop the machinery of (strict) fans and apply it to detecting the $k$-space and Ascoli properties in spaces that naturally appear in General Topology, Functional Analysis, and Topological Algebra. In particular, we detect (generalized) metric spaces $X$ whose functor-spaces, functions spaces, free (para)Topological (abelian) groups, free (locally convex) linear Topological spaces, free (Lawson) Topological semilattices, and free (para)Topological (Clifford, Abelian) inverse semigroups are $k$-spaces or Ascoli spaces.
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some open problems in Topological Algebra
arXiv: General Topology, 2012Co-Authors: Taras Banakh, Mitrofan M Choban, Igor Guran, Igor ProtasovAbstract:This is the list of open problems in Topological Algebra posed on the conference dedicated to the 20th anniversary of the Chair of Algebra and Topology of Lviv National University, that was held on 28 September 2001.
Bikchentaev A. - One of the best experts on this subject based on the ideXlab platform.
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The continuity of multiplication for two topologies associated with a semifinite trace on von neumann Algebra
2020Co-Authors: Bikchentaev A.Abstract: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) < ∞
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Local convergence in measure on semifinite von Neumann Algebras
2020Co-Authors: Bikchentaev A.Abstract:Suppose that M is a von Neumann Algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. The set M̃ of all τ-measurable operators with the topology tτ of convergence in measure is a Topological *-Algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing t τ and are denoted by tτ1 and twτ1, respectively. The set M̃ with any of these topologies is a Topological vector space. The continuity of certain operations and the closedness of certain classes of operators in M̃ with respect to the topologies t τ1 and twτ1 are proved. S.M. Nikol'skii's theorem (1943) is extended from the Algebra B(H) to semifinite von Neumann Algebras. The following theorem is proved: For a von Neumann Algebra M with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the Algebra M is finite; (ii) twτ1 = tτ1; (iii) the multiplication is jointly tτ1-continuous from M̃ × M̃ to M̃; (iv) the multiplication is jointly twτ1- continuous from M̃ × M̃ to M̃; (v) the involution is t τ1-continuous from M̃ to M̃. © 2006 Pleiades Publishing, Inc
Arkady Leiderman - One of the best experts on this subject based on the ideXlab platform.
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omega omega dominated function spaces and omega omega bases in free objects of Topological Algebra
arXiv: General Topology, 2016Co-Authors: Taras Banakh, Arkady LeidermanAbstract:A Topological space $X$ is defined to have an $\omega^\omega$-base if at each point $x\in X$ the space $X$ has a neighborhood base $(U_\alpha[x])_{\alpha\in\omega^\omega}$ such that $U_\beta[x]\subset U_\alpha[x]$ for all $\alpha\le\beta$ in $\omega^\omega$. We characterize Topological and uniform spaces whose free (locally convex) Topological vector spaces or free (Abelian or Boolean) Topological groups have $\omega^\omega$-bases.
Tadesse Bekeshie - One of the best experts on this subject based on the ideXlab platform.
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two fixed point theorems in Topological vector space valued cone metric spaces with complete Topological Algebra cones
IOSR Journal of Mathematics, 2012Co-Authors: Tadesse BekeshieAbstract:In this paper we prove two fixed point theorems in Topological vector space valued cone metric spaces (briefly TVS-CMS). To that end we introduce the concept of complete Topological Algebra cone (briefly CTA cone). Our theorems are generalizations of corresponding theorems in (8) and (13). The paper also gives answers to the open problems posed in (15). Finally we give an application of our first theorem. AMS Mathematics Subject Classification (2010): 47H10, 54H25, 46A40 Key terms: Topological vector space, ordered Topological vector space, Algebra over a field, Topological Algebra over a field, Topological vector space valued cone metric space, scalarization function, CTA cone
Vladimir Pestov - One of the best experts on this subject based on the ideXlab platform.
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Universal arrows to forgetful functors from categories of Topological Algebra
Bulletin of the Australian Mathematical Society, 1993Co-Authors: Vladimir PestovAbstract:We survey the present trends in theory of universal arrows to forgetful functors from various categories of Topological Algebra and functional analysis to categories of topology and Topological Algebra. Among them are free Topological groups, free locally convex spaces, free Banach-Lie Algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.
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Universal arrows to forgetful functors from categories of Topological Algebra
arXiv: Functional Analysis, 1992Co-Authors: Vladimir PestovAbstract:We survey the present trends in theory of universal arrows to forgetful functors from various categories of Topological Algebra and functional analysis to categories of topology and Topological Algebra. Among them are free Topological groups, free locally convex spaces, free Banach-Lie Algebras, and much more. An accent is put on relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and others may hopefully become amenable to methods of this theory.