Maps preserving quadratic product between positive cones of JBW-algebras

ABSTRACT

Let $\mathcal{A}$ be a JBW-algebra. For $a\in \mathcal{A}$, let $U_a$ denote the quadratic operator defined by $U_a\left(b\right)=2a\circ (a\circ b)-a^2\circ b$ $(b\in \mathcal{A})$. We show that if $\mathcal{A}$ has no abelian direct summand, then every bijection φ from the positive cone of $\mathcal{A}$ onto the positive cone of a JBW-algebra $\mathcal{B}$ satisfying $\phi \left(U_{a}b\right)=U_{\phi \left(a\right)}\phi \left(b\right)$ for all positive elements $a,b\in \mathcal{A}$ extends to a Jordan isomorphism from $\mathcal{A}$ onto $\mathcal{B}$.

Keywords:

• JBW-algebra
• quadratic operator
• quadratic product
• preserver mapping

Acknowledgments

The authors express their sincere thanks to the reviewers for their careful reading and their valuable comments and suggestions towards to the improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.