Let A be a Banach algebra admitting a bounded approximate unit and satisfying property \(\mathbb {B}\) . Suppose \(T: A \rightarrow X\) 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 \(\Rightarrow T(b)\cdot a + b\cdot T(a) =0\) );
(ii) There exist an element \(\xi \in X^{**}\) and a linear map (actually a bounded Jordan derivation) \(d: A\rightarrow X\) satisfying \(\xi \cdot a = a \cdot \xi \in X\) , \(T(a) = d(a) +\xi \cdot a\) , and \(d(b)\cdot a + b\cdot d(a)= - 2 \xi \cdot (b a),\) for all \(a,b\in A\) with \(a b =0\) .
Assuming that A is a \(\hbox {C}^*\) -algebra, we show that a bounded linear mapping \(T: A\rightarrow X\) is anti-derivable at zero if, and only if, there exist an element \(\eta \in X^{**}\) and an anti-derivation \(d: A \rightarrow X\) satisfying \(\eta \cdot a = a \cdot \eta \in X\) , \(\eta \cdot [a,b] = 0\) (i.e., \(L_{\eta }: A \rightarrow A\) , \(L_{\eta } (a) = \eta \cdot a\) vanishes on commutators), and \(T(a) = d(a) +\eta \cdot a\) , for all \(a,b \in A\) . The results are also applied for some special operator algebras.
