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## Studia Mathematica

2008 | 184 | 1 | 31-47
Tytuł artykułu

### Jordan isomorphisms and maps preserving spectra of certain operator products

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Let 𝓐₁, 𝓐₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁ ∗ ⋯ ∗ T_{k}$ on elements in $𝓐_{i}$, which covers the usual product $T₁ ∗ ⋯ ∗ T_{k} = T₁ ⋯ T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: 𝓐₁ → 𝓐₂ be a (not necessarily linear) map satisfying $σ(Φ(A₁) ∗ ⋯ ∗ Φ(A_{k})) = σ(A₁ ∗ ⋯ ∗ A_{k})$ whenever any one of $A_{i}$'s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.
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Tom
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31-47
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wydano
2008
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autor
• Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P.R. of China
autor
• Department of Mathematics, The College of William & Mary, Williamsburg, VA 13185, U.S.A.
autor
• Department of Applied Mathematics, National Sun Yat-sen University, and National Center for Theoretical Sciences, Kaohsiung 80424, Taiwan
• Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
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