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Characterizations of Lie derivations of triangular algebras. (English) Zbl 1226.16026
A triangular algebra $$T=T(A,X,B)$$ has the form of an upper triangular matrix ring with elements having diagonal entries in $$A$$ and $$B$$ and upper right entries in $$X$$; $$A$$ and $$B$$ are unital algebras over a commutative ring $$R$$ with 1, and $$X$$ is an $$A$$-$$B$$-bimodule that is faithful on each side. The center of $$T$$ is $$Z(T)=\{\text{diag}(a,b)\in T\mid am=mb$$ for all $$m\in X\}$$. The projection of $$Z(T)$$ to either $$A$$ or $$B$$ is central.
The main results in the paper assume that these projections are all of $$Z(A)$$ and $$Z(B)$$. The first theorem considers an $$R$$-linear map $$\delta$$ of $$T$$ satisfying $$\delta([x,y])=[\delta(x),y]+[x,\delta(y)]$$ for all $$x,y\in T$$ so that $$xy=0$$ and proves that on $$T$$, $$\delta(x)=d(x)+\tau(x)$$ for $$d$$ a derivation of $$T$$ and $$\tau$$ an $$R$$-linear map from $$T$$ to $$Z(T)$$ with $$\tau([x,y])=0$$ when $$xy=0$$. The second main result is the first with the assumptions that $$xy=0$$ replaced with $$xy=\text{diag}(1_A,0_B)$$, but also requires the additional assumption that for any $$a\in A$$ there is an integer $$j$$ so that $$j1_A-a$$ is invertible.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16S50 Endomorphism rings; matrix rings 47B47 Commutators, derivations, elementary operators, etc.
##### Keywords:
Lie derivations; triangular algebras; linear maps; commutators
Full Text:
##### References:
 [1] Alaminos, J.; Extremera, J.; Villena, A.R.; Bresar, 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] An, R.; Hou, J.C., Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear algebra appl., 431, 1070-1080, (2009) · Zbl 1173.47023 [4] Bresar, M., Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. roy. soc. Edinburgh sect. A, 137, 9-21, (2007) · Zbl 1130.16018 [5] 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 [6] Cheng, W.S., Commuting maps of triangular algebras, J. London math. soc., 63, 117-127, (2001) · Zbl 1014.16035 [7] Cheng, W.S., Lie derivations of triangular algebras, Linear and multilinear algebra, 51, 299-310, (2003) · Zbl 1060.16033 [8] Christensen, E., Derivations of nest algebras, Math. ann., 229, 155-161, (1977) · Zbl 0356.46057 [9] K.R. Davision, Nest Algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific and Technical, Burnt Mill Harlow, Essex, UK, 1988. [10] 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 [11] Jiao, M.; Hou, J.C., Additive maps derivable at zero point on nest algebras, Linear algebra appl., 432, 2984-2994, (2010) · Zbl 1191.47047 [12] Jing, W., On Jordan all-derivable points of $$B(H)$$, Linear algebra appl., 430, 941-946, (2009) · Zbl 1163.47030 [13] Jing, W.; Lu, S.; Li, P., Characterisations of derivations on some operator algebras, Bull. austral. math. soc., 66, 227-232, (2002) · Zbl 1035.47019 [14] Lu, F.Y.; Jing, W., Characterizations of Lie derivations of $$B(X)$$, Linear algebra appl., 432, 89-99, (2010) · Zbl 1188.47029 [15] Qi, X.F.; Hou, J.C., Characterizations of derivations of Banach space nest algebras: all-derivable points, Linear algebra appl., 432, 3183-3200, (2010) · Zbl 1192.47071 [16] Zhu, J., All-derivable points of operator algebras, Linear algebra appl., 427, 1-5, (2007) · Zbl 1128.47062 [17] Zhu, J.; Xiong, C., Derivable mappings at unit operator on nest algebras, Linear algebra appl., 422, 721-735, (2007) · Zbl 1140.47059 [18] Zhu, J.; Xiong, C.P., All-derivable points in continuous nest algebras, J. math. anal. appl., 340, 845-853, (2008) · Zbl 1134.47054
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