# Quaternions and Functional Calculus

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Basics of C*-Algebras, or Why Are C*-Algebras Beautiful?

**Theorem**

**1**

**Definition**

**1.**

**Definition**

**2.**

- X is a $\mathbb{H}$-vector space or quaternionic two sided vector space if X is a vector space with respect to the right scalar multiplication $X\times \mathbb{H}\ni (x,q)\mapsto xq\in X$ and is also a vector space with respect to the left scalar multiplication $\mathbb{H}\times X\ni (q,x)\mapsto qx\in X$.
- Let X be a $\mathbb{H}$-vector space, then X is called a $\mathbb{H}$-algebra if the following conditions hold:$$q\left(xy\right)=\left(qx\right)y,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(xq\right)y=x\left(qy\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left(xy\right)q=x\left(yq\right)$$
- A $\mathbb{H}$-involution on X is a self-map $a\mapsto {a}^{*}$ of a $\mathbb{H}$-algebra X such that$${x}^{**}=x,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\left(qx\right)}^{*}={x}^{*}{q}^{*},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\left(xq\right)}^{*}={q}^{*}{x}^{*}$$for every $x,y\in X$ and $q\in \mathbb{H}$. Then X is called a $\mathbb{H}$-involutive algebra.
- Let X be a $\mathbb{H}$-involutive algebra that is equipped with a $\mathbb{H}$-norm $\parallel .\parallel $. Then X is called a $\mathbb{H}$-${C}^{*}$-algebra if it is complete under the norm and satisfies$$1+{x}^{*}xisinvertibleintheunitizationofAforanyelementx\in Aand\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\parallel {x}^{*}{x\parallel =\parallel x\parallel}^{2}.$$

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Gelfand Theory with Generalized Characters

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**2.**

**Lemma**

**1.**

- The characters have the same kernel.
- The characters are equivalent in the sense of Definition 6.
- The characters are equal when restricted to the real part subalgebra ${A}_{R}$ of $A.$

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. S-Functional Calculus

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 5. Fueter Regular Functions

**Definition**

**11.**

**Lemma**

**2**

**(Poincaré–Stokes**

**lemma).**

**Definition**

**12.**

## 6. Structural Regular Functions

## 7. Cauchy’s Theorem and Integral Formula

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

## 8. A New Analogue of the Cauchy Integral Formula

**Proposition**

**3.**

## 9. S-Resolvent Operator and S-Spectrum

**Theorem**

**7.**

**Proof.**

**Definition**

**13**

**Definition**

**14**

## 10. Functional Calculus

**Theorem**

**8.**

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Ghamari, E.; Kučerovský, D.
Quaternions and Functional Calculus. *Symmetry* **2019**, *11*, 953.
https://doi.org/10.3390/sym11080953

**AMA Style**

Ghamari E, Kučerovský D.
Quaternions and Functional Calculus. *Symmetry*. 2019; 11(8):953.
https://doi.org/10.3390/sym11080953

**Chicago/Turabian Style**

Ghamari, Elham, and Dan Kučerovský.
2019. "Quaternions and Functional Calculus" *Symmetry* 11, no. 8: 953.
https://doi.org/10.3390/sym11080953