Narrow Results

$({\phi }_{+},{\phi }_{-})$-derivations defined from a Banach–Jordan pair $V=(V^{+},V^{-})$ into a semisimple Banach–Jordan pair $W=(W^{+},W^{-})$ are automatically continuous provided that $\phi =({\phi }_{+},{\phi }_{-})$$({\phi }_{\sigma }:V^{\sigma }\to W^{\sigma })$ is an epimorphism.

The main objective of this research paper consists in introducing the concept of $({\psi }_{+},{\psi }_{-})$-derivations acting on Banach–Jordan pairs and Banach–Jordan algebras and giving an automatic continuity result of the operators in question under some algebraic conditions. Concretely, we prove that $({\psi }_{+},{\psi }_{-})$ -derivations defined from a Banach–Jordan pair $V=(V^{+},V^{-})$ into a strongly prime Banach–Jordan pair $W=(W^{+},W^{-})$  are automatically continuous under the assumptions that the socle $\mathrm{\text{Soc}}(W)=(\mathrm{\text{Soc}}(W^{+}),\mathrm{\text{Soc}}(W^{-}))$  is nonzero and $\psi =({\psi }_{+},{\psi }_{-})$  is an isomorphism from $V$  into $W$. Similar results for Banach–Jordan algebras hold to be true.