Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-11T07:29:34.247Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizing Jordan maps on C*-algebras through zero products

Part of: Special classes of linear operators Selfadjoint operator algebras

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
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

References

1. Alaminos, J., Brešar, M., Extremera, J. and Villena, A. R., Characterizing homomorphisms and derivations on C*-algebras, Proc. R. Soc. Edinb. A 137 (2007), 1–7.CrossRefGoogle Scholar
2. Alaminos, J., Brešar, M., Extremera, J. and Villena, A. R., Maps preserving zero products, Studia Math. 193 (2009), 131–159.CrossRefGoogle Scholar
3. Ara, P. and Mathieu, M., Local multipliers of C*-algebras, Springer Monographs in Mathematics (Springer, 2003).CrossRefGoogle Scholar
4. Brešar, M., Jordan mappings of semiprime rings, J. Alg. 127 (1989), 218–228.CrossRefGoogle Scholar
5. Bre?sar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Am. Math. Soc. 335 (1993), 525–546.CrossRefGoogle Scholar
6. Brešar, M. and Šemrl, P., On bilinear maps on matrices with applications to commutativity preservers, J. Alg. 301 (1989), 803–837.CrossRefGoogle Scholar
7. Brešar, M., Chebotar, M.A. and Martindale III, W. S., Functional identities, Frontiers in Mathematics (Birkhäuser, 2007).Google Scholar
8. Burgos, M., Fernández-Polo, F. J., J. Garcés, J., Martínez Moreno, J. and Peralta, A. M., Orthogonality preservers in C*-algebras, JB*-algebras and JB*-triples, J. Math. Analysis Applic. 348 (2008), 220–233.CrossRefGoogle Scholar
9. A. Chebotar, M., Ke, W.-F. and Lee, P.-H., Maps preserving zero Jordan products on Hermitian operators, Illinois J. Math. 49 (2) (2006), 445–452.Google Scholar
10. A. Chebotar, M., Ke, W.-F., Lee, P.-H. and Zhang, R., On maps preserving zero Jordan products, Monatsh. Math. 149 (2) (2006), 91–101.CrossRefGoogle Scholar
11. G. Dales, H., Banach algebras and automatic continuity, London Mathematical Society Monographs (New Series), Volume 24 (Oxford Scientific Publications/Clarendon Press/ Oxford University Press, New York, 2000).Google Scholar
12. Hadwin, D. and Li, J., Local derivations and local automorphisms, J. Math. Analysis Applic. 290 (2004), 702–714.CrossRefGoogle Scholar
13. Herstein, I. N., Jordan homomorphisms, Trans. Am. Math. Soc. 81 (1956), 331–351.CrossRefGoogle Scholar
14. Hou, J. and Zhao, L., Zero-product preserving additive maps on symmetric operator spaces and self-adjoint operator spaces, Linear Alg. Applic. 399 (2005), 235–244.CrossRefGoogle Scholar
15. Hou, J. and Zhao, L., Jordan zero-product preserving additive maps on operator algebras, J. Math. Analysis Applic. 314 (2006), 689–700.Google Scholar
16. E. Johnson, B., Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Camb. Phil. Soc. 120 (1996), 455–473.CrossRefGoogle Scholar
17. Omladič, M., Radjavi, H., and Šemrl, P., Preserving commutativity, J. Pure Appl. Alg. 156 (2001), 309–328.CrossRefGoogle Scholar