*-Jordan Semi-Triple Derivable Mappings
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DOI: 10.1007/s13226-020-0434-4
∗-JORDAN SEMI-TRIPLE DERIVABLE MAPPINGS Lin Chen∗,∗∗ and Jianhua Zhang∗ ∗ College
of Mathematics and Information Science, Shaanxi Normal University, Xian, 710062, China
∗∗ Department
of Mathematics and Physics, Anshun University, Anshun 561000 China
e-mails: [email protected]; [email protected] (Received 16 October 2018; after final revision 9 April 2019; accepted 25 April 2019) In this paper, we characterize the ∗-Jordan semi-triple derivable mappings (i.e. a mapping Φ from ∗ algebra A into A satisfying Φ(AB ∗ A) = Φ(A)B ∗ A + AΦ(B)∗ A + AB ∗ Φ(A) for every A, B ∈ A) in the finite dimensional case and infinite dimensional case. Key words : Jordan semi-triple derivable mapping; derivation; matrix algebra. 2010 Mathematics Subject Classification : 47B49, 46K15.
1. I NTRODUCTION It is a surprising result of Martindale [16] that every multiplicative bijective mapping from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive. This result was ˇ utilized by Semrl in [20] to describe the form of the semigroup isomorphisms of standard operator algebras on Banach spaces. Some other results on the additivity of multiplicative mappings between operator algebras can be found in [3, 5, 13, 14, 17, 18]. Besides additivity of multiplicative mappings, additivity of derivable mappings is also an interesting problem. Let A be an algebra. Recall that a mapping φ from A into A is called a derivable mapping if φ(AB) = φ(A)B + Aφ(B) for all A, B ∈ A and a Jordan derivable mapping if φ(AB + BA) = φ(A)B + Aφ(B) + φ(B)A + Bφ(A) for all A, B ∈ A. We say that additive derivable mappings are additive derivations, and additive Jordan derivable mappings are additive Jordan derivations. Lu [15] showed that each Jordan derivable mapping of a 2-torsion free prime ring containing a nontrivial
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idempotent is also additive. Let [A, B] = AB − BA be the usual Lie product of A and B. Recall that a mapping φ from A into A is called a Lie derivable mapping if φ([A, B]) = [φ(A), B] + [A, φ(B)] for all A, B ∈ A. Lu [12] gave a characterization of Lie derivable mapping of operator algebra on Banach space. Some other results on derivable mappings can be found in [1, 7, 11]. In recent years, the additivity of *-derivable mappings has attracted the attentions of many researchers. Let A be an algebra with involution, a mapping φ : A → A is called a ∗-Lie derivable mapping if for any A, B ∈ A, φ([A, B]∗ ) = [φ(A), B]∗ + [A, φ(B)]∗ , where [A, B]∗ = AB − BA∗ is the skew Lie product of A and B. In [21] Yu and Zhang showed that every ∗-Lie derivable mapping from a factor von Neumann algebra on an infinite dimensional complex Hilbert space into itself is an additive ∗-derivation. In [10], Li, Lu and Fang arrived the same conclusion on von Neumann algebra without central abelian projections. Jing [8] proved that every ∗-Lie derivable mapping of standard operator algebra on complex Hilbert space is an inner ∗-derivation. A mapping φ : A → A is called a
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