On a dilation of a some class of completely positive maps

In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert $C^*$-module $\mathcal{M}$ over $C^*$-algebra $B$ and taking values in $B.$ The set of all such defined sesquilinear forms is denoted by $\mathcal{S}_{B}(\mathcal{M}).$ We consider completely positive maps from locally $C^*$-algebra $A$ to $\mathcal{S}_{B}(\mathcal{M}).$ Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.