Let \( \mathcal {L} \) be a \( \mathcal {J} \) -subspace lattice on Banach space X and \( \textrm{Alg} \mathcal {L} \) be the associated \( \mathcal {J} \) -subspace lattice algebra. A ternary derivation on \( \textrm{Alg} \mathcal {L} \) is defined as a triple of linear mappings \( (\gamma , \delta , \tau ): \textrm{Alg} \mathcal {L} \rightarrow \textrm{Alg} \mathcal {L} \) that satisfies the condition \( \gamma (A B) = \delta (A) B + A \tau (B) \) for all \( A, B \in \textrm{Alg} \mathcal {L} \) . We establish that for given linear maps \(\delta , \tau \) on \( \textrm{Alg} \mathcal {L} \) , there exists a unique linear map \( \gamma \) on \( \textrm{Alg} \mathcal {L} \) defined by \( \gamma (A) = R A + AS \) for some \( R, S \in L (X) \) such that \( (\gamma , \delta , \tau )\) forms a ternary derivation on \( \textrm{Alg} \mathcal {L} \) if and only if \( \delta , \tau \) satisfy \( \delta (A) B + A \tau (B) = 0 \) for any \( A, B \in \textrm{Alg} \mathcal {L} \) with \( AB =0 \) . As applications of this result, we provide a comprehensive characterization of linear mappings \( \delta \) and \( \tau \) . Additionally, we investigate linear mappings that are derivable at zero, (left/right) centralizers, (left/right) ideal-preserving mappings, and local (generalized) derivations within JSL algebras. These findings are applicable to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras in Banach spaces.
