Inverse-producing extensions of Topological Algebras

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Algebra extension of containing an inverse element to a given


The course covers a basic concept of considering mathematical properties in extensions of a given topological algebra. In doing so, we extends a algebra to an extension containing , checking a property of an element in the extension . In this course we treat multiplicative invertibility as a mathematical property and consider, among other things, topological properties that allow to create an algebra extension in which an multiplicative inverse of a given element exists.

is satisfied and is the one-element of the multiplication. Essentially, this involves topological properties of the element that either allows invertibility in a particular extension of or never has an inverse element in any extensions of , i.e., is permanently singular. The basic sets with a multiplicative linkage here are topological algebras, where the linkages are.

  • (TA1) multiplication of a vector by a scalar as an outer linkage,
  • (TA2) addition of vectors in the vector space as an inner linkage, and
  • (TA3) multiplication of two vectors as inner linkage

are continuous in each case. Here, a vector space with properties (TA1) and (TA2) is called a topological vector space. If there is additionally a multiplication is additionally this multiplicative inner linkage continuous (TA3) then the vector space is called a topological algebra.

Origine of the Course[edit | edit source]

This course was created in the german Wikiversity for a lecture with Wiki2Reveal slides, that can be annotated in the browser (no online storage of annotation). The course material will be translated as Open Educational Resources that can be maintained and updated in Wikiversity. The concept of inverse-producing algebra extensions is based on Richard Arens work on that topic for normed algebras[1]. Other classes of topological algebras were considered by Wieslaw Zelazko like locally convex and multiplicative locally convex algebras[2]. The course includes

Wiki2Reveal Contents[edit | edit source]

Chapter 0: Preliminaries[edit | edit source]

Chapter 1: Introduction into topological algebras[edit | edit source]

Chapter 2: K-singular elements[edit | edit source]

First, we shall discuss topological criteria which ensure that an element is permanently singular in any algebra extension of class . If the negation of the topological property causes the element to have an inverse element in an algebra extension of class , a topoligical invertibility criterion arises.

Chapter 3: K-regular elements[edit | edit source]

In this chapter, given topological criteria, we construct algebra extensions algebra extensions of class in which a given -regular element is invertible.

(3.1) Normed and locally bounded algebras[edit | edit source]

(3.2) Multiplicative topological algebras[edit | edit source]

(3.3) Locally convex and locally pseudoconvex algebras[edit | edit source]

(3.4) Topological algebras[edit | edit source]

Chapter 4: Solvability of equations[edit | edit source]

In this chapter, the invertibility with is considered as a special case of the solvability of an equation with . Here, an algebra expansion of is used to search for a solution that solves the equation .

Use of materials for lectures[edit | edit source]

The lecture is provided in a PanDoc slide format (PanDocElectron-SLIDE) in Wikiversity, which can be transferred to annotatable slides using the Wiki2Reveal tool or using [PanDocElectron to load the Wikiversity source available online and convert it to presentation slides that can be used offline. You can also use Wiki2Reveal to create a RevealJS or DZSlides presentation directly from Wikiversity articles].

Origin of materials[edit | edit source]

In the spirit of OER (Open Educational Resources), the lecture content should be made freely available. Initially, the slides created from the customizable wiki content were made available in a GitHib repository to facilitate download and use. However, maintaining and updating the content in a repository is not very efficient due to the fact that any update in Wikiversity requires also an update of the slides in the repository. Therefore, Wiki2Reveal was developed for the lecture slides, which allows to generate lecture slides directly from the Wikiversity content and to annotate the slides online as well. The Wikiversity slides are represented as sections and usually any annotation is performed either directly on the slide by pressing (C) for comment. Furthermore the lecturer is able to use a separate whiteboard for every slide by pressing (B) on the slide an return to the slide by pressing (B) again. Annotation mode for the slides can be switched by pressing (C) again. Keep the sections of Wiki2Reveal articles small so that the generated slides fit on the screen. More detailed text about the slides is usually created in separate articles. If the explanation pages explicitly refer to a slide, the explanation page gets a marker PanDocElectron-TEXT and SLIDE or TEXT version refer to each other reciprocally.

See also[edit | edit source]

Wiki Books[edit | edit source]

References[edit | edit source]

  1. Arens Richard, (1958), Inverse producing extensions of normed algebras, Trans. Amer. Math. Soc. 88, S. 536-548
  2. Zelazko Wieslaw, (1971), Selected topics in topological algebras, Aarhus University lecture notes, No. 31