# An algebraic semi-automated proof of the fundamental identity of Jordan algebras

@article{Wetering2018AnAS, title={An algebraic semi-automated proof of the fundamental identity of Jordan algebras}, author={John van de Wetering}, journal={arXiv: Rings and Algebras}, year={2018} }

The fundamental identity of quadratic Jordan algebras $Q_{Q_a b} = Q_aQ_bQ_a$ is commonly proven as a consequence of MacDonalds theorem or using more analytic methods. In this short note we give a self-contained purely algebraic proof using just a few easily proven identities and a Python script that follows a simple randomised logic to reduce expressions of Jordan operators.

## 3 Citations

Pure Maps between Euclidean Jordan Algebras

- MathematicsElectronic Proceedings in Theoretical Computer Science
- 2019

We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of Kraus rank one channels ($A\mapsto B^*AB$). We show that this…

Commutativity in Jordan operator algebras

- MathematicsJournal of Pure and Applied Algebra
- 2020

Abstract While Jordan algebras are commutative, operator commutativity will behave badly for a general non-associative Jordan algebra. When the product operators of two elements commute, the elements…

Quantum Theory from Principles, Quantum Software from Diagrams

- Physics
- 2021

This thesis consists of two parts. The first part is about how quantum theory can be recovered from first principles, while the second part is about the application of diagrammatic reasoning,…

## References

SHOWING 1-5 OF 5 REFERENCES

A Taste of Jordan Algebras

- Mathematics
- 2003

In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim…

Geometry Of State Spaces Of Operator Algebras

- Mathematics
- 2002

Preface * JB-Algebras * JBW-Algebras * Structure of JBW-Algebras * Representations of JB-Algebras * State Spaces of Jordan Algebras * Dynamical Correspondences * General Compressions * Spectral…

Analysis on Symmetric Cones

- Mathematics
- 1995

I. Convex cones II. Jordan algebras III. Symmetric cones and Euclidean Jordan algebras IV. The Peirce decomposition in a Jordan algebra V. Classification of Euclidean Jordan algebras VI. Polar…

Jordan Structures in Geometry and Analysis

- Mathematics
- 2012

Preface 1. Jordan and Lie theory 2. Jordan structures in geometry 3. Jordan structures in analysis Bibliography Index.

Størmer. Jordan operator algebras, volume 21

- Pitman Advanced Pub. Program,
- 1984