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CHARACTERIZATIONS OF HIGHER DERIVATIONS AND JORDAN HIGHER DERIVATIONS ON CSL ALGEBRAS

Part of: Special classes of linear operators

Published online by Cambridge University Press:  15 March 2011

JIANKUI LI  and
JIANBIN GUO
JIANKUI LI*
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China (email: jiankuili@yahoo.com)
JIANBIN GUO
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China (email: jianbin-guo@163.com)
*
✉For correspondence; e-mail: jiankuili@yahoo.com
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Abstract

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Let ℒ be a commutative subspace lattice and 𝒜=Alg ℒ. It is shown that every Jordan higher derivation from 𝒜 into itself is a higher derivation. We say that D=(δi)i∈ℕ is a higher derivable linear mapping at G if δn(AB)=∑ i+j=nδi(A)δj(B) for all n∈ℕ and A,B∈𝒜 with AB=G. We also prove that if D=(δi)i∈ℕ is a bounded higher derivable linear mapping at 0 from 𝒜 into itself and δn (I)=0 for all n≥1 , or D=(δi)i∈ℕ is a higher derivable linear mapping at I from 𝒜 into itself, then D=(δi)i∈ℕ is a higher derivation.

Keywords

CSL algebrahigher derivable linear mappinghigher derivationJordan higher derivation

MSC classification

Secondary: 47B47: Commutators, derivations, elementary operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 486 - 499
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This work is supported by NSF of China.

References

[1]Alaminos, J., Extremera, J., Villena, A. and Brešar, M., ‘Characterizing homomorphisms and derivations on C *-algebras’, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1–7.CrossRefGoogle Scholar
[2]Brešar, M., ‘Jordan derivations on semiprime rings’, Proc. Amer. Math. Soc. 104 (1988), 1003–1006.CrossRefGoogle Scholar
[3]Chebotar, M., Ke, W. and Lee, P., ‘Maps characterized by action on zero products’, Pacific J. Math. 216 (2004), 217–228.CrossRefGoogle Scholar
[4]Crist, R., ‘Local derivations on operator algebras’, J. Funct. Anal. 135 (1996), 76–92.CrossRefGoogle Scholar
[5]Davidson, K., Nest Algebras, Pitman Research Notes in Mathematics Series, 191 (Longman Scientific, Harlow, 1988).Google Scholar
[6]Ferrero, M. and Haetinger, C., ‘Higher derivations and a theorem by Herstein’, Quaest. Math. 25 (2002), 249–257.CrossRefGoogle Scholar
[7]Herstein, I., ‘Jordan derivations of prime rings’, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.CrossRefGoogle Scholar
[8]Jing, W., Lu, S. and Li, P., ‘Characterisations of derivations on some operator algebras’, Bull. Aust. Math. Soc. 66 (2002), 227–232.CrossRefGoogle Scholar
[9]Johnson, B., ‘Local derivations on C *-algebras are derivations’, Trans. Amer. Math. Soc. 353 (2001), 313–325.CrossRefGoogle Scholar
[10]Kadison, R., ‘Local derivations’, J. Algebra 130 (1990), 494–509.CrossRefGoogle Scholar
[11]Li, J. and Pan, Z., ‘Annihilator-preserving maps, multipliers, and derivations’, Linear Algebra Appl. 432 (2010), 5–13.CrossRefGoogle Scholar
[12]Lu, F., ‘Characterizations of derivations and Jordan derivations on Banach algebras’, Linear Algebra Appl. 430 (2009), 2233–2239.CrossRefGoogle Scholar
[13]Lu, F., ‘The Jordan structure of CSL algebras’, Studia Math. 190 (2009), 283–299.CrossRefGoogle Scholar
[14]Nakajima, A., ‘On generalized higher derivations’, Turkish J. Math. 24 (2000), 295–311.Google Scholar
[15]Xiao, Z. and Wei, F., ‘Jordan higher derivations on triangular algebras’, Linear Algebra Appl. 432 (2010), 2615–2622.CrossRefGoogle Scholar