On commuting $U$-operators in Jordan algebras

TL;DR

This paper investigates the conditions under which $U$-operators in Jordan algebras commute, showing that non-degeneracy is essential and providing a simpler proof for linear Jordan algebras in characteristic not 2.

Contribution

It demonstrates that the commutativity of $U$-operators does not hold in general without non-degeneracy and offers a simplified proof for linear Jordan algebras.

Findings

Non-degeneracy is necessary for the commutativity of $U$-operators.

A simpler proof exists for linear Jordan algebras when char F ≠ 2.

The original result is not universally valid without additional assumptions.

Abstract

Recently J.A.Anquela, T.Cort\'es, and H.Petersson proved that for elements $x, y$ in a non-degenerate Jordan algebra $J$, the relation $x \circ y = 0$ implies that the $U$-operators of $x$ and $y$ commute: $U_xU_y = U_yU_x$. We show that the result may be not true without the assumption on non-degeneracity of $J$. We give also a more simple proof of the mentioned result in the case of linear Jordan algebras, that is, when $char\, F\neq 2$.