# Inverse-producing extensions of Topological Algebras

Algebra extension ${\displaystyle B}$ of ${\displaystyle A}$ containing an inverse element ${\displaystyle b=z^{-1}\in B}$ to a given ${\displaystyle z\in A}$

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The course covers a basic concept of considering mathematical properties in extensions of a given topological algebra. In doing so, we extends a algebra ${\displaystyle A}$ to an extension ${\displaystyle B}$ containing ${\displaystyle A\subset B}$, checking a property of an element ${\displaystyle z\in A}$ in the extension ${\displaystyle z\in B}$. 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 ${\displaystyle B}$ in which an multiplicative inverse of a given element ${\displaystyle z\in A}$ exists.

${\displaystyle \exists _{z^{-1}\in B}:\,z\cdot z^{-1}=z^{-1}\cdot z=e}$

is satisfied and ${\displaystyle e\in A}$ is the one-element of the multiplication. Essentially, this involves topological properties of the element ${\displaystyle z\in A}$ that either allows invertibility in a particular extension ${\displaystyle B}$ of ${\displaystyle A}$ or never has an inverse element in any extensions ${\displaystyle B}$ of ${\displaystyle A}$, 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

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

### Chapter 2: K-singular elements

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

### Chapter 3: K-regular elements

In this chapter, given topological criteria, we construct algebra extensions algebra extensions of class ${\displaystyle {\mathcal {K}}}$ in which a given ${\displaystyle {\mathcal {K}}}$-regular element is invertible.

### Chapter 4: Solvability of equations

In this chapter, the invertibility ${\displaystyle z\cdot x=x\cdot z=e}$ with ${\displaystyle x=z^{-1}}$ is considered as a special case of the solvability of an equation ${\displaystyle z_{1}\cdot x=z_{2}}$ with ${\displaystyle z_{1},z_{2}\in A}$. Here, an algebra expansion ${\displaystyle B}$ of ${\displaystyle A}$ is used to search for a solution ${\displaystyle x\in B}$ that solves the equation ${\displaystyle z_{1}\cdot x=z_{2}}$.

## Use of materials for lectures

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

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.