Maps Preserving the Spectrum of Generalized Jordan Product of Operators
نویسنده
چکیده
Let A1,A2 be standard operator algebras on complex Banach spaces X1, X2, respectively. For k ≥ 2, let (i1, . . . , im) be a sequence with terms chosen from {1, . . . , k}, and define the generalized Jordan product T1 ◦ · · · ◦ Tk = Ti1 · · ·Tim + Tim · · ·Ti1 on elements in Ai. This includes the usual Jordan product A1 ◦ A2 = A1A2 + A2A1, and the triple {A1, A2, A3} = A1A2A3 + A3A2A1. Assume that at least one of the terms in (i1, . . . , im) appears exactly once. Let a map Φ : A1 → A2 satisfy that σ(Φ(A1) ◦ · · · ◦ Φ(Ak)) = σ(A1 ◦ · · · ◦Ak) whenever any one of A1, . . . , Ak 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 mth root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.
منابع مشابه
On strongly Jordan zero-product preserving maps
In this paper, we give a characterization of strongly Jordan zero-product preserving maps on normed algebras as a generalization of Jordan zero-product preserving maps. In this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly Jordan zero-product preserving maps are completely different. Also, we prove that the direct p...
متن کاملAdditive Maps Preserving Idempotency of Products or Jordan Products of Operators
Let $mathcal{H}$ and $mathcal{K}$ be infinite dimensional Hilbert spaces, while $mathcal{B(H)}$ and $mathcal{B(K)}$ denote the algebras of all linear bounded operators on $mathcal{H}$ and $mathcal{K}$, respectively. We characterize the forms of additive mappings from $mathcal{B(H)}$ into $mathcal{B(K)}$ that preserve the nonzero idempotency of either Jordan products of operators or usual produc...
متن کاملA Note on Spectrum Preserving Additive Maps on C*-Algebras
Mathieu and Ruddy proved that if be a unital spectral isometry from a unital C*-algebra Aonto a unital type I C*-algebra B whose primitive ideal space is Hausdorff and totallydisconnected, then is Jordan isomorphism. The aim of this note is to show that if be asurjective spectrum preserving additive map, then is a Jordan isomorphism without the extraassumption totally disconnected.
متن کاملThe second dual of strongly zero-product preserving maps
The notion of strongly Lie zero-product preserving maps on normed algebras as a generalization of Lie zero-product preserving maps are dened. We give a necessary and sufficient condition from which a linear map between normed algebras to be strongly Lie zero-product preserving. Also some hereditary properties of strongly Lie zero-product preserving maps are presented. Finally the second dual of...
متن کاملSpectrum Preserving Linear Maps Between Banach Algebras
In this paper we show that if A is a unital Banach algebra and B is a purely innite C*-algebra such that has a non-zero commutative maximal ideal and $phi:A rightarrow B$ is a unital surjective spectrum preserving linear map. Then $phi$ is a Jordan homomorphism.
متن کامل