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Characterizations of Lie derivations of \(B(X)\). (English) Zbl 1188.47029
The authors study a variant of the concept of a Lie derivation in the setting of bounded linear operators on a Banach space \(X\) of dimension at least \(3\). The two results proven in this paper state that, if \(\delta: B(X)\to B(X)\) is a linear mapping satisfying the Lie derivation property on commutators \([a,b]\) with (i) \(ab=0\) or (ii) \(ab\) is a fixed non-trivial idempotent \(p\), then \(\delta\) is the sum of a derivation \(d\) on \(B(X)\) and a linear centre-valued mapping \(\tau\) vanishing on commutators \([a,b]\) satisfying (i) or (ii), respectively.

MSC:
47B47 Commutators, derivations, elementary operators, etc.
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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[1] Alaminos, J.; Extremera, J.; Villena, A.R.; Bres̆ar, M., Characterizing homomorphisms and derivations on \(C^\ast\)-algebras, Proc. roy. soc. Edinburgh sect. A, 137, 1-7, (2007) · Zbl 1144.47030
[2] Alaminos, J.; Mathieu, M.; Villena, A.R., Symmetric amenability and Lie derivations, Math. proc. Cambridge philos. soc., 137, 433-439, (2004) · Zbl 1063.46033
[3] Bres̆ar, M., Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. roy. soc. Edinburgh sect. A, 137, 9-21, (2007) · Zbl 1130.16018
[4] Chebotar, M.A.; Ke, W.-F.; Lee, P.-H., Maps characterized by action on zero products, Pacific J. math., 216, 217-228, (2004) · Zbl 1078.16034
[5] Crist, R.L., Local derivations on operator algebras, J. funct. anal., 135, 76-92, (1996) · Zbl 0902.46046
[6] Han, D., The first cohomology groups of nest algebras on normed spaces, Proc. amer. math. soc., 118, 1147-1149, (1993) · Zbl 0802.47044
[7] Hou, J.C.; Qi, X.F., Additive maps derivable at some points on J-subspace lattice algebras, Linear algebra appl., 429, 1851-1863, (2008) · Zbl 1153.47062
[8] Jing, W., On Jordan all-derivable points of \(B(H)\), Linear algebra appl., 430, 941-946, (2009) · Zbl 1163.47030
[9] Jing, W.; Lu, S.; Li, P., Characterisations of derivations on some operator algebras, Bull. austral. math. soc., 66, 2, 227-232, (2002) · Zbl 1035.47019
[10] Johnson, B.E., Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. proc. Cambridge philos. soc., 120, 455-473, (1996) · Zbl 0888.46024
[11] Johnson, B.E., Local derivations on \(C^\ast\)-algebras are derivations, Trans. amer. math. soc., 353, 313-325, (2001) · Zbl 0971.46043
[12] Kadison, R.V., Local derivations, J. algebra, 130, 494-509, (1990) · Zbl 0751.46041
[13] Larson, D.R.; Sourour, A.R., Local derivations and local automorphisms of \(B(X)\), Proc. sym. pure math., 51, 187-194, (1990)
[14] Lu, F., Lie derivations of \(\mathcal{J}\)-subspace lattice algebras, Proc. amer. math. soc., 135, 2581-2590, (2007) · Zbl 1116.47060
[15] Lu, F., Lie derivations of certain CSL algebras, Israel J. math., 155, 149-156, (2006) · Zbl 1130.47055
[16] Mathieu, M.; Villena, A.R., The structure of Lie derivations on \(C^\ast\)-algebras, J. funct. anal., 202, 504-525, (2003) · Zbl 1032.46086
[17] Miers, C.R., Lie derivations of von Neumann algebras, Duke math. J., 40, 403-409, (1973) · Zbl 0264.46064
[18] Shulman, V.S., Operators preserving ideals in \(C^\ast\)-algebras, Studia math., 109, 67-72, (1994) · Zbl 0821.46085
[19] Zhu, J., All-derivable points of operator algebras, Linear algebra appl., 427, 1-5, (2007) · Zbl 1128.47062
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