Jordan triple product homomorphisms on Hermitian matrices to and from dimension one

TL;DR

This paper characterizes all Jordan triple product homomorphisms involving Hermitian matrices and scalar fields, providing a complete description of such mappings in various contexts.

Contribution

It provides a comprehensive characterization of Jordan triple product homomorphisms on Hermitian matrices and from scalar fields to Hermitian matrices, extending previous understanding.

Findings

Explicit forms of all such homomorphisms are derived.

Results apply to mappings from Hermitian matrices to complex numbers and vice versa.

The characterizations include cases over real and nonnegative real numbers.

Abstract

We characterise all Jordan triple product homomorphisms, that is, mappings $\Phi$ satisfying $$ \Phi(ABA) = \Phi(A)\Phi(B)\Phi(A) $$ from the set of all Hermitian $n \times n$ complex matrices to the field of complex numbers. Further we characterise all Jordan triple product homomorphisms from the field of complex or real numbers or the set of all nonnegative real numbers to the set of all Hermitian $n \times n$ complex matrices.