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Let A1 and A2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H1 and H2, respectively. For k≥2, let (i1,…,im) be a fixed sequence with i1,…,im∈{1,…,k} and assume that at least one of the terms in (i1,…,im) appears exactly once. Define the generalized Jordan product T1∘T2∘⋯∘Tk=Ti1Ti2⋯Tim+Tim⋯Ti2Ti1 on elements in Ai. Let Φ:A1→A2 be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that Φ satisfies that σπ(Φ(A1)∘⋯∘Φ(Ak))=σπ(A1∘⋯∘Ak) for all A1,…,Ak, where σπ(A) stands for the peripheral spectrum of A, if and only if there exist a scalar c∈{-1,1} and a unitary operator U:H1→H2 such that Φ(A)=cUAU* for all A∈A1, or Φ(A)=cUAtU* for all A∈A1, where At is the transpose of A for an arbitrarily fixed orthonormal basis of H1. Moreover, c=1 whenever m is odd.

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