Orthogonally $a$-Jensen mappings on $C^*$-modules

Abstract

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