On commuting $U$-operators in Jordan algebras

Abstract

Recently J.A.Anquela, T.Cortés, 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: $U_x{U}_y=U_y{U}_x$. 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 $charF\ne 2$.