Inverse-producing extensions of Topological Algebras/Algebra of polynomials

Introduction

A polynomial algebra ${\displaystyle A[t]}$ is a vector space of polynomials, where the coefficients come from the given algebra ${\displaystyle A}$. The polynomial algebra ${\displaystyle A[t]}$ is an essential tool to construct an algebra extension ${\displaystyle B}$ of ${\displaystyle A}$ in which a given ${\displaystyle z\in A}$ is invertible if it satisfies certain topological invertibility criteria.

Remark: Algebra expansion

In the construction of algebra extension in which a ${\displaystyle z\in A}$ is invertible, the first step is to consider the algebra of polynomials ${\displaystyle A[t]}$. The following figure shows how the algebra extension ${\displaystyle B}$ is constructed over the polynomial algebra.

Algebraic closure

We only extend the algebra ${\displaystyle A}$ to contain an additional element ${\displaystyle t\notin A}$, which is to be contained in an algebra extension ${\displaystyle B}$ of ${\displaystyle A}$. Since multiplication and addition must be completed in ${\displaystyle B}$, polynomials result from multiplications ${\displaystyle a\cdot t}$ and ${\displaystyle t\cdot t=t^{2}}$ with coefficients ${\displaystyle a\in A}$n, which must be contained as summands ${\displaystyle a_{n}\cdot t^{n}}$ as polynomials in the algebra expansion.

Extension of the algebra

This implies the closure of the

• multiplicative linkage of ${\displaystyle t}$ with itself and therefore ${\displaystyle t^{n}}$ with ${\displaystyle n\in \mathbb {N} }$ must also be in ${\displaystyle B}$ again,
• the arbitrary multiplicative links of ${\displaystyle t^{n}\in B}$ with elements from ${\displaystyle A}$ lie again in ${\displaystyle B}$, i.e. ${\displaystyle a_{n}\cdot t^{n}\in B}$.
• the additive algebraic algebraic closure also eventually requires that additive links from ${\displaystyle a_{n}\cdot t^{n}\in B}$ lie again in ${\displaystyle B}$.

Polynomials with coefficients from the given algebra

Out of this necessity, one considers polynomials with coefficients from ${\displaystyle A}$ as the first step in constructing an algebra expansion in which a ${\displaystyle z}$ can be invertible.

Polynomial algebra

We now consider, for a given topological algebra ${\textstyle \left(A,\left\|\cdot \right\|_{\mathcal {A}}\right)\in {\mathcal {K}}(\mathbb {K} )}$, the set of polynomials with coefficients in ${\displaystyle A}$.

${\displaystyle p(t)=\sum _{k=0}^{n}p_{k}\cdot t^{k}{\mbox{ with }}p_{k}\in A{\mbox{ for }}k\in \{0,1,...,n\}}$

and power series with coefficients in algebra ${\displaystyle A}$.

${\displaystyle p(t)=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ with }}p_{k}\in A{\mbox{ for }}k\in \mathbb {N} _{o}}$

Degree of polynomials

First of all, polynomials would be more formally notated in the above form with ${\displaystyle n\in \mathbb {N} _{o}}$ and with ${\displaystyle p_{n}\not =0_{A}}$ ${\displaystyle n}$ would indicate the degree of the polynomial. For the Cauchy product of two polynomials ${\displaystyle p,q\in A[t]}$, however, this notation is unsuitable, since in the addition and multiplication two polynomials ${\displaystyle p,q\in A[t]}$ the handling of the degree entails additional formal overhead, which, however, does not matter for the further considerations of algebra expansions.

Notation for the polynomial algebra

Therefore, polynomials are defined as follows over "finite" sequences ${\displaystyle c_{oo}(A)}$, which from an index bound ${\displaystyle n\in \mathbb {N} _{o}}$ consists only of the zero vector ${\displaystyle 0_{A}}$ in ${\displaystyle A}$.

${\displaystyle p(t)=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ with }}(p_{k})_{k\in \mathbb {N} _{0}}\in c_{oo}(A)}$

Cauchy product

Given, in general, two polynomials ${\displaystyle p,q}$ with coefficients from ${\textstyle A}$.

${\displaystyle p(t):=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ und }}q(t):=\sum _{k=0}^{\infty }q_{k}\cdot t^{k}}$

Then Cauchy product of ${\displaystyle p,q}$ is defined as follows:

${\displaystyle p(t)\cdot q(t):=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{n}p_{k}\cdot q_{n-k}\right)t^{n}.}$