2025/12/10 by Li, Jiankui, Peralta, Antonio M., Su, Shanshan
#46L05 #46L57 #FOS: Mathematics #Operator Algebras (math.OA) #Primary 47B47 #Secondary 47B48
paper · doi:10.48550/arxiv.2512.09578
Let A be a Banach algebra admitting a bounded approximate unit and satisfying property \mathbbB. Suppose 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 in A ⇒ T(b)⋅ a + b⋅ T(a) =0); (ii) There exist an element ξ∈ X** and a linear map (actually a bounded Jordan derivation) d: A→ X satisfying ξ⋅ a = a ⋅ ξ∈ X, T(a) = d(a) +ξ⋅ a, and d(b)⋅ a + b⋅ d(a)= - 2 ξ⋅ (b a), for all a,b∈ A with a b =0. Assuming that A is a C^*-algebra we show that a bounded linear mapping T: A→ X is anti-derivable at zero if, and only if, there exist an element η∈ X** and an anti-derivation d: A → X satisfying η⋅ a = a ⋅ η∈ X, η⋅ [a,b] = 0 \rm(i.e., Lη: A → A, Lη (a) = η⋅ a vanishes on commutators\rm), and T(a) = d(a) +η⋅ a, for all a,b ∈ A. The results are also applied for some special operator algebras.