Maps preserving peripheral spectrum of generalized Jordan products of operators

TL;DR

This paper characterizes maps between operator algebras that preserve the peripheral spectrum of generalized Jordan products, showing such maps are essentially Jordan isomorphisms scaled by roots of unity.

Contribution

It provides a complete characterization of spectrum-preserving maps for generalized Jordan products on operator algebras, extending previous results to more complex products.

Findings

Maps preserving peripheral spectrum are Jordan isomorphisms times roots of unity.

The result applies to generalized Jordan products including the usual Jordan and triple products.

The characterization holds for maps with range containing all operators of rank at most three.

Abstract

Let $X_1$ and $X_2$ be complex Banach spaces with dimension at least three, $\mathcal{A}_1$ and $\mathcal{A}_2$ be standard operator algebras on $X_1$ and $X_2$, respectively. For $k\geq2$, let $(i_1,...,i_m)$ be a sequence with terms chosen from $\{1,\ldots,k\}$ and assume that at least one of the terms in $(i_1,\ldots,i_m)$ appears exactly once. Define the generalized Jordan product $T_1\circ T_2\circ\cdots\circ T_k=T_{i_1} T_{i_2}\cdots T_{i_m}+T_{i_m}\cdots T_{i_2} T_{i_1}$ on elements in $\mathcal{A}_i$. This includes the usual Jordan product $A_1A_2+A_2A_1$, and the Jordan triple $A_1A_2A_3+A_3A_2A_1$. Let $\Phi:\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a map with range containing all operators of rank at most three. It is shown that $\Phi$ satisfies that $\sigma_\pi(\Phi(A_1)\circ\cdots\circ\Phi(A_k))=\sigma_\pi(A_1\circ\cdots\circ A_k)$ for all $A_1, \ldots, A_k$, where $\sigma_\pi(A)$ stands for the peripheral spectrum of $A$, if and only if $\Phi$ is a Jordan isomorphism multiplied by an $m$th root of unity.