Characterizing Jordan maps on C-algebras through zero products

Let A and B be C*-algebras, let X be an essential Banach A-bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T(a)T(b) + T(b)T(a) = 0 and S(a)b + bS(a) + aS(b) + S(b)a = 0 whenever a, b ε A are such that ab = ba = 0. We prove that then T = wΦ and S = D...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 2010-10, Vol.53 (3), p.543-555
Main Authors: Alaminos, J., Brešar, J. M., Extremera, J., Villena, A. R.
Format: Article
Language:English
Subjects:
46L05
Algebra
C-algebra
derivation
homomorphism
Homomorphisms
Jordan derivation
Jordan homomorphism
Mathematical analysis
Primary 47B48
zero-product-preserving map
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Summary:Let A and B be C*-algebras, let X be an essential Banach A-bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T(a)T(b) + T(b)T(a) = 0 and S(a)b + bS(a) + aS(b) + S(b)a = 0 whenever a, b ε A are such that ab = ba = 0. We prove that then T = wΦ and S = D + Ψ, where w lies in the centre of the multiplier algebra of B, Φ: A → B is a Jordan epimorphism, D: A → X is a derivation and Ψ: A → X is a bimodule homomorphism.
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091509000534