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.

In this paper, we give a complete description of the structure of separating linear maps between continuous fields of Banach spaces. Some automatic continuity results are obtained.

This paper has two main objectives. One of them is to obtain some conditions under which the image of a $\varphi$-derivation on a Banach algebra ${𝒜}$ is contained in the Jacobson radical of ${𝒜}$ and other purpose of this study is to present some results about the automatic continuity of $\varphi$-derivations in Banach algebras.

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.

We study the relationship between the existence of nonprincipal ultrafilters over ω and the failure of the automatic continuity, Steinhaus and Bergman properties for infinite products of finite groups.