## Abstract

Let ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces ${H}_{1}$ and ${H}_{2}$, respectively. For $k\ge 2$, let $({i}_{1},\dots ,{i}_{m})$ be a fixed sequence with ${i}_{1},\dots ,{i}_{m}\in $ $\{1,\dots ,k\}$ and assume that at least one of the terms in $({i}_{1},\dots ,{i}_{m})$ 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 ${\mathcal{A}}_{i}$. Let $\mathrm{\Phi}:{\mathcal{A}}_{1}\to {\mathcal{A}}_{2}$ be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that $\mathrm{\Phi}$ satisfies that ${\sigma}_{\pi}(\mathrm{\Phi}({A}_{1})\circ \cdots \circ \mathrm{\Phi}({A}_{k}))={\sigma}_{\pi}({A}_{1}\circ \cdots \circ {A}_{k})$ for all ${A}_{1},\dots ,{A}_{k}$, where ${\sigma}_{\pi}(A)$ stands for the peripheral spectrum of $A$, if and only if there exist a scalar $c\in \{-\mathrm{1,1}\}$ and a unitary operator $U:{H}_{1}\to {H}_{2}$ such that $\mathrm{\Phi}(A)=cUA{U}^{\mathrm{*}}$ for all $A\in {\mathcal{A}}_{1}$, or $\mathrm{\Phi}(A)=cU{A}^{t}{U}^{\mathrm{*}}$ for all $A\in {\mathcal{A}}_{1}$, where ${A}^{t}$ is the transpose of $A$ for an arbitrarily fixed orthonormal basis of ${H}_{1}$. Moreover, $c=1$ whenever $m$ is odd.

## Citation

Wen Zhang. Jinchuan Hou. "Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators." Abstr. Appl. Anal. 2014 1 - 8, 2014. https://doi.org/10.1155/2014/192040