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Characterizing Jordan maps on C*-algebras through zero products

Part of: Selfadjoint operator algebras Special classes of linear operators

Published online by Cambridge University Press:  05 August 2010

J. Alaminos ,
J. M. Brešar ,
J. Extremera  and
A. R. Villena
J. Alaminos
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; jlizana@ugr.es; avillena@ugr.es)
J. M. Brešar
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, 2000 Maribor, Slovenia (matej.bresar@fmf.uni-lj.si)
J. Extremera
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; jlizana@ugr.es; avillena@ugr.es)
A. R. Villena
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; jlizana@ugr.es; avillena@ugr.es)
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Abstract

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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.

Keywords

C*-algebrahomomorphismJordan homomorphismderivationJordan derivationzero-product-preserving map

MSC classification

Secondary: 47B48: Operators on Banach algebras 46L05: General theory of $C^*$-algebras

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 543 - 555
Copyright
Copyright © Edinburgh Mathematical Society 2010

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