Orthogonally $a$-Jensen mappings on $C^{\ast }$-modules

Abstract

We investigate the representation of the so-called orthogonally $a$-Jensen mappings acting on $C^{\ast }$-modules. More precisely, let $\mathfrak{A}$ be a unital $C^{\ast }$-algebra with the unit $1$, let $a\in \mathfrak{A}$ be fixed such that $a,1-a$ are invertible and let $\mathcal{E},\mathcal{F},\mathcal{G}$ be inner product $\mathfrak{A}$-modules. We prove that if there exist additive mappings $\phi ,\psi$ from $\mathcal{F}$ into $\mathcal{E}$ such that $⟨\phi (y),\psi (z)⟩=0$ and $a⟨\phi (y),\phi (z)⟩a^{\ast }=(1-a)⟨\psi (y),\psi (z)⟩(1-a{)}^{\ast }$ for all $y,z\in \mathcal{F}$, then a mapping $f:\mathcal{E}\to \mathcal{G}$ is orthogonally $a$-Jensen if and only if it is of the form $f(x)=A(x)+B(x,x)+f(0)$ for $x\in \mathcal{K}:=\phi (\mathcal{F})+\psi (\mathcal{F})$, where $A:\mathcal{E}\to \mathcal{G}$ is an $a$-additive mapping on $\mathcal{K}$ and $B$ is a symmetric $a$-biadditive orthogonality preserving mapping on $\mathcal{K}\times \mathcal{K}$. Some other related results are also presented.