TY - GEN
T1 - Local and 2-local Lie-type Derivations of Operator Algebras on Banach Spaces
AU - Deng, Zhi Cheng
AU - Wei, Feng
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024
Y1 - 2024
N2 - Let X be a Banach space over the field F (F is either the real field R or the complex field C). Let B(X) be the set of all bounded linear operators on X and F(X) be the set of all finite rank operators in B(X). A subalgebra A of B(X) is called a standard operator algebra if A contains F(X). Suppose that δ is a map from A into B(X). Firstly, we prove that if δ is a Lie-type derivation, then δ has the standard form. Furthermore, we show that if δ is a local Lie-type derivation, then δ is a Lie-type derivation. Finally, we prove that if δ is a 2-local Lie n-derivation, then δ=d+τ, where d is a derivation, and τ is homogeneous map from A into FI such that τ(A+B)=τ(A) for each A, B in A where B is a sum of (n-1)-th commutators.
AB - Let X be a Banach space over the field F (F is either the real field R or the complex field C). Let B(X) be the set of all bounded linear operators on X and F(X) be the set of all finite rank operators in B(X). A subalgebra A of B(X) is called a standard operator algebra if A contains F(X). Suppose that δ is a map from A into B(X). Firstly, we prove that if δ is a Lie-type derivation, then δ has the standard form. Furthermore, we show that if δ is a local Lie-type derivation, then δ is a Lie-type derivation. Finally, we prove that if δ is a 2-local Lie n-derivation, then δ=d+τ, where d is a derivation, and τ is homogeneous map from A into FI such that τ(A+B)=τ(A) for each A, B in A where B is a sum of (n-1)-th commutators.
KW - 2-local Lie-type derivation
KW - Algebra of bounded linear operators
KW - Local Lie-type derivation
KW - Standard operator algebra
UR - http://www.scopus.com/inward/record.url?scp=85193572172&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-50795-3_13
DO - 10.1007/978-3-031-50795-3_13
M3 - Conference contribution
AN - SCOPUS:85193572172
SN - 9783031507946
T3 - Springer Proceedings in Mathematics and Statistics
SP - 175
EP - 188
BT - Advances in Ring Theory and Applications - WARA22
A2 - Ali, Shakir
A2 - Ashraf, Mohammad
A2 - Rehman, Nadeem ur
A2 - De Filippis, Vincenzo
PB - Springer
T2 - Workshop on Associative Rings and Algebras with Additional Structures, WARA 2022
Y2 - 18 July 2022 through 20 July 2022
ER -