$\varphi$-maps on Hilbert $C^*$-modules, $\varphi$-module domains and ternary domains

For an operator-valued $\varphi$-map $\Phi$ on a Hilbert $C^*$-module $X$ over a $C^*$-algebra $A$; $X_\Phi = \{x\in X;\Phi(xa) = \Phi(x)\varphi(a) $ for all $ a\in A\}$ is its $\varphi$-module domain and $T_\Phi = \{x\in X;\Phi(y \langle x, z\rangle) = \Phi(y)\Phi(x)^* \Phi(z) $ for all $ y, z \in X\}$ is its ternary domain. In this talk we will discuss about some properties of $X_\Phi$ and $T_\Phi$.
This is a joint work with M.B. Asadi and R. Behmani.