On the ternary domain of a completely positive map on a Hilbert C*-module

We associate to an operator valued completely positive linear map $\phi$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{\phi }$ of $X,$ called ‘$\mathit{\text{ternary}}\mathit{\text{domain}}$’ of $\phi$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $\phi$ and every $\phi$-map (i.e., associated quaternary map with $\phi$) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $\phi$ on A\ is a closed two-sided $\ast$-ideal $T_{\phi }$ of the multiplicative domain of $\phi$. We show that $X{T}_{\phi }=X_{\phi }$ and give several characterizations of the set $X_{\phi }\mathrm{.}$ Furthermore, we establish some relationships between $X_{\phi }$ and minimal Stinespring dilation triples associate to $\phi$. Finally, we show that every operator valued completely positive linear map $\phi$ on a $C^{\ast }$ -algebra $A$ induces a unique (in a some sense) completely positive linear map on the linking algebra of $X$ and we determine its multiplicative domain in terms of the multiplicative domain of $\phi$ and the ternary domain of $\phi$ on $X$.