Maps preserving the spectrum of generalized Jordan product of operators

By J. Hou, Chi-Kwong Li, Ngai-Ching Wong. 2010

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Abstract Let A 1 , A 2 be standard operator algebras on complex Banach spaces X 1 , X 2 , respectively. For k ⩾ 2 , let ( i 1 , … , i m ) be a sequence with terms chosen from { 1 , … , k } , and define the generalized Jordan product T 1 ∘ ⋯ ∘ T k = T i 1 ⋯ T i m + T i m ⋯ T i 1 on elements in A i . This includes the usual Jordan product A 1 ∘ A 2 = A 1 A 2 + A 2 A 1 , and the triple { A 1 , A 2 , A 3 } = A 1 A 2 A 3 + A 3 A 2 A 1 . Assume that at least one of the terms in ( i 1 , … , i m ) appears exactly once. Let a map Φ : A 1 → A 2 satisfy that σ ( Φ ( A 1 ) ∘ ⋯ ∘ Φ ( A k ) ) = σ ( A 1 ∘ ⋯ ∘ A k ) whenever any one of A 1 , … , A k has rank at most one. It is shown in this paper that if the range of Φ contains all operators of rank at most three, then Φ must be a Jordan isomorphism multiplied by an m th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/63fc0deb635d54931e8af663777cc90d983922b9