  We associate to an operator valued completely positive linear map $\varphi$
on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a
subset $X_{\varphi }$ of $X,$ called '\textit{ternary domain}' of $\varphi$ on
$X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of
$\varphi $ and every $\varphi $-map (i.e., associated quaternary map with
$\varphi $) acts on it as a ternary map. We also provide several
characterizations for this set. The ternary domain \ of $\varphi $ on $A\ $ is
a closed two-sided $\ast $-ideal $T_{\varphi }$ of the multiplicative domain of
$\varphi $. We show that $XT_{\varphi }=X_{\varphi }$ and give several
characterizations of the set $X_{\varphi }.$ Furthermore, we establish some
relationships between $X_{\varphi }$ and minimal Stinespring dilation triples
associate to $\varphi $. Finally, we show that every operator valued completely
positive linear map $\varphi $ 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 $\varphi $ and the ternary domain of $\varphi $ on $X$.
