Normalizing rings

García-Pacheco, Francisco Javier; Sáez-Martínez, Sol

doi:10.1007/s43037-020-00055-0

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…Another concept on which we will strongly rely is the one of practical ring [33] (Definition 6). A topological ring R is said to be practical provided that 0 ∈ cl(U(R)), that is, 0 belongs to the closure of the invertibles U(R) of R. Practical rings serve to extend the classical Operator Theory to the topological module setting.…”

Section: Methodsmentioning

confidence: 99%

“…A topological ring R is said to be practical provided that 0 ∈ cl(U(R)), that is, 0 belongs to the closure of the invertibles U(R) of R. Practical rings serve to extend the classical Operator Theory to the topological module setting. An extensive study on practical topological rings can be found in [33].…”

Section: Methodsmentioning

confidence: 99%

“…Next, let us assume that R is a totally right-feasible topological division ring. According to [33] (Lemma 2), ker( f ) is a maximal submodule of M. Therefore, cl(ker( f )) is either ker( f ) or M. If cl(ker( f )) = ker( f ), then ker( f ) is closed in M, hence f is continuous in view of [33] (Lemma 3), which is a contradiction. As a consequence, M = cl(ker( f )).…”

Section: Example 2 Letmentioning

confidence: 97%

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Chaos in Topological Modules

García-Pacheco

2022

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Chaotic and pathological phenomena in topological modules are studied in this manuscript. In particular, constructions of noncontinuous linear functionals are provided for a wide variety of topological modules. In addition, constructions of balanced and absorbing sets which are not neighborhoods of zero are also given in an extensive class of topological modules. Finally, we construct a linearly open set with empty interior in a large amount of topological modules. All these constructions are related to each other. Prior to developing all these results, we provide an axiomatization of the topological concept of limit by introducing the limit operators in a similar context as hull operators or closure operators are defined.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Example 2 Letmentioning

confidence: 97%

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Chaos in Topological Modules

García-Pacheco

2022

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“…We refer the reader to [17,18] for a wider perspective on topological rings and modules. In [19,Theorem (1)], it was proved that, for any topological module M, N 0 (M) := V ∈N 0 (M) V is a closed submodule of M whose inherited topology is the trivial topology.…”

Section: Topological Backgroundmentioning

confidence: 99%

“…for all k ∈ N. A topological ring R is called practical provided that 0 ∈ cl(U(R)). We refer the reader to [19,[22][23][24][25] for a wider perspective on practical rings. Every normed algebra is trivially a practical ring.…”

Section: Invertibility In Topological Ringsmentioning

confidence: 99%

Invertibles in topological rings: a new approach

García-Pacheco

Miralles

Murillo‐Arcila

2021

RACSAM

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Every element in the boundary of the group of invertibles of a Banach algebra is a topological zero divisor. We extend this result to the scope of topological rings. In particular, we define a new class of semi-normed rings, called almost absolutely semi-normed rings, which strictly includes the class of absolutely semi-valued rings, and prove that every element in the boundary of the group of invertibles of a complete almost absolutely semi-normed ring is a topological zero divisor. To achieve all these, we have to previously entail an exhaustive study of topological divisors of zero in topological rings. In addition, it is also well known that the group of invertibles is open and the inversion map is continuous and C-differentiable in a Banach algebra. We also extend these results to the setting of complete normed rings. Finally, this study allows us to generalize the point, continuous and residual spectra to the scope of Banach algebras.

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Topological Modules Geometry

García-Pacheco

2023

J Math Sci

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The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital $$C^*$$ C ∗ -algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to $$C^*$$ C ∗ -algebras.

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Normalizing rings

Cited by 9 publications

References 14 publications

Chaos in Topological Modules

Chaos in Topological Modules

Invertibles in topological rings: a new approach

Topological Modules Geometry

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