Linear Maps Which are Anti-derivable at Zero
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Linear Maps Which are Anti-derivable at Zero Doha Adel Abulhamil1 · Fatmah B. Jamjoom1 · Antonio M. Peralta2 Received: 11 November 2019 / Revised: 16 February 2020 © The Author(s) 2020
Abstract Let T : A → X be a bounded linear operator, where A is a C∗ -algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., ab = 0 in A implies T (b)a + bT (a) = 0); (b) There exist an anti-derivation d : A → X ∗∗ and an element ξ ∈ X ∗∗ satisfying ξ a = aξ, ξ [a, b] = 0, T (ab) = bT (a) + T (b)a − bξ a, and T (a) = d(a) + ξ a, for all a, b ∈ A. We also prove a similar equivalence when X is replaced with A∗∗ . This provides a complete characterization of those bounded linear maps from A into X or into A∗∗ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are ∗ -anti-derivable at zero. Keywords C∗ -algebra · Banach bimodule · Derivation · Anti-derivation · Maps anti-derivable at zero · Maps ∗ -anti-derivable at zero Mathematics Subject Classification Primary 46L05 · 46L57 · 47B47; Secondary 15A86
Communicated by Mohammad Sal Moslehian.
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Antonio M. Peralta [email protected] Doha Adel Abulhamil [email protected] Fatmah B. Jamjoom [email protected]
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Present Address: Mathematics Departments, College of Sciences, King Abdulaziz University, P.O. Box 9039, Jeddah 21413, Saudi Arabia
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Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
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D. A. Abulhamil et al.
1 Introduction Let us begin this note by formulating a typical problem in recent studies about preservers. Suppose X is a Banach A-bimodule over a complex Banach algebra A. A derivation from A to X is a linear mapping D : A → X satisfying the following algebraic identity: D(ab) = D(a)b + a D(b), ∀(a, b) ∈ A2 .
(1.1)
A derivation D is called inner if there exists x0 ∈ X such that D(a) = δx0 (a) = [a, x0 ] = ax0 − x0 a for all a ∈ A. A typical challenge on preservers can be posed in the following terms: Problem 1 Suppose T : A → X is a linear map satisfying (1.1) only on a proper subset D ⊂ A2 . Is T a derivation? There is no need to comment that the role of the set D is the real core of the question. A typical example is provided by the set Dz := {(a, b) ∈ A2 : ab = z}, where z is a fixed point in A. A linear map T : A → X is said to be a derivation at a point z ∈ A if the identity (1.1) holds for every (a, b) ∈ Dz . In the literature, a linear map which is a derivation at a point z is also called derivable at z. Let us point out that there exist linear maps which are derivable at zero, but they are not derivations (for example, the identity mapping on a complex Banach algebra is a derivation at zero, but it is not a derivation). If T : A → B is a linear mapping from A into another Banach algebra satisfying T (ab) = T (a)T (b) for all (a, b) ∈ Dz , we say that T is a homomorphism at the point z. Linear maps which are Jordan (∗ -)derivations, o
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