New Insights Into Linear Maps Which are Anti-Derivable at Zero
Abstract
Abstract Let A be a Banach algebra admitting a bounded approximate unit and satisfying property $$\mathbb {B}$$ B . Suppose $$T: A \rightarrow X$$ T : A → X is a continuous linear map, where X is an essential Banach A -bimodule. We prove that the following statements are equivalent: ( i ) T is anti-derivable at zero (i.e., $$a b =0$$ a b = 0 in A $$\Rightarrow T(b)\cdot a + b\cdot T(a) =0$$ ⇒ T ( b ) · a + b · T ( a ) = 0 ); ( ii ) There exist an element $$\xi \in X^{**}$$ ξ ∈ X ∗ ∗ and a linear map (actually a bounded Jordan derivation) $$d: A\rightarrow X$$ d : A → X satisfying $$\xi \cdot a = a \cdot \xi \in X$$ ξ · a = a · ξ ∈ X , $$T(a) = d(a) +\xi \cdot a$$ T ( a ) = d ( a ) + ξ · a , and $$d(b)\cdot a + b\cdot d(a)= - 2 \xi \cdot (b a),$$ d ( b ) · a + b · d
Fields of Study & Subjects
Advanced Topics in Algebra
Functional Equations Stability Results
Advanced Operator Algebra Research
Publication details
- Journal / source
- Bulletin of the Malaysian Mathematical Sciences Society
- Publisher
- Springer Science and Business Media LLC
- Volume
- 49
- Issue
- 3
- Publication date
- 6 May 2026
- Language
- EN