Jordan triple product homomorphisms on Hermitian matrices of dimension two

TL;DR

This paper characterizes all mappings on 2x2 Hermitian matrices that preserve the Jordan triple product structure, providing a complete description of such homomorphisms.

Contribution

It offers a complete characterization of Jordan triple product homomorphisms specifically for 2x2 Hermitian matrices, a case not fully understood before.

Findings

All such homomorphisms are explicitly described.

The structure of these homomorphisms is fully determined.

The results extend understanding of Jordan triple products in low dimensions.

Abstract

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$ \Phi(ABA) = \Phi(A)\Phi(B)\Phi(A) $$ on the set of all Hermitian $2 \times 2$ complex matrices.