Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators

Let and be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces and , respectively. For , let be a fixed sequence with and assume that at least one of the terms in appears exactly once. Define the generalized Jordan product on elements in . Let be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that satisfies that for all , where stands for the peripheral spectrum of , if and only if there exist a scalar and a unitary operator such that for all , or for all , where is the transpose of for an arbitrarily fixed orthonormal basis of . Moreover, whenever is odd.