Left derivable maps at non-trivial idempotents on nest algebras

Hoger Ghahramani; Saman Sattari

Published: 2019-04-05
Language: EN

Vol. 33 (2019)

Left derivable maps at non-trivial idempotents on nest algebras

Hoger Ghahramani , Saman Sattari

Abstract

Let Alg???? be a nest algebra associated with the nest ???? on a (real or complex) Banach space ????. Suppose that there exists a non-trivial idempotent P∈Alg???? with range P(????)∈????, and δ:Alg????→Alg???? is a continuous linear mapping (generalized) left derivable at P, i.e. δ(ab) = aδ(b) + bδ(a) (δ(ab) = aδ(b) + bδ(a) - baδ(I)) for any a,b∈Alg???? with ab = P, where I is the identity element of Alg????. We show that δ is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg???? with the property that δ(P) = 2Pδ(P) or δ(P) = 2Pδ(P) - Pδ(I) for every idempotent P in Alg????.

(EN)

Keywords:

nest algebra , left derivable , left derivation (EN)

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Ghahramani, H., & Sattari, S. (2019). Left derivable maps at non-trivial idempotents on nest algebras. Annales Mathematicae Silesianae, 33, 97–105. Retrieved from https://trrest.vot.pl/ojsus/index.php/AMSIL/article/view/13653

References

1. F.F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin, 1973.
2. M. Brešar, Characterizations of derivations on some normed algebras with involution, J. Algebra 152 (1992), 454–462.
3. M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh. Sect. A 137 (2007), 9–21.
4. M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), 7–16.
5. K.R. Davidson, Nest Algebras, Pitman Res. Notes in Math., vol. 191, Longman, London, 1988.
6. B. Fadaee and H. Ghahramani, Jordan left derivations at the idempotent elements on reflexive algebras, Publ. Math. Debrecen 92 (2018), 261–275.
7. H. Ghahramani, Additive mappings derivable at non-trivial idempotents on Banach algebras, Linear Multilinear Algebra 60 (2012), 725–742.
8. H. Ghahramani, On centralizers of Banach algebras, Bull. Malays. Math. Sci. Soc. 38 (2015), 155–164.
9. H. Ghahramani, Characterizing Jordan maps on triangular rings through commutative zero products, Mediterr. J. Math. 15 (2018), Art. 38, 10 pp., DOI: 10.1007/s00009-018-1082-3.
10. N.M. Ghosseiri, On Jordan left derivations and generalized Jordan left derivations of matrix rings, Bull. Iranian Math. Soc. 38 (2012), 689–698.
11. J.C. Hou and X.L. Zhang, Ring isomorphisms and linear or additive maps preserving zero products on nest algebras, Linear Algebra Appl. 387 (2004), 343–360.
12. J.K. Li and J. Zhou, Jordan left derivations and some left derivable maps, Oper. Matrices 4 (2010), 127–138.
13. N.K. Spanoudakis, Generalizations of certain nest algebra results, Proc. Amer. Math. Soc. 115 (1992), 711–723.
14. J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequationes Math. 75 (2008), 260–266.
15. B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (1991), 609–614.
16. J. Zhu and C.P. Xiong, Derivable mappings at unit operator on nest algebras, Linear Algebra Appl. 422 (2007), 721–735. Google Scholar

Hoger Ghahramani

h.ghahramani@uok.ac.ir

Affiliation:

Department of Mathematics, University of Kurdistan, Iran Iran, Islamic Republic of

Saman Sattari

Affiliation:

Department of Mathematics, University of Kurdistan, Iran Iran, Islamic Republic of

Vol. 33 (2019)
Published: 2019-07-18

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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