  In this article, we investigate certain basic properties of invariant
multilinear CP maps. For instance, we prove Russo-Dye type theorem for
invariant multilinear positive maps on both commutative $C^*$-algebras and
finite-dimensional $C^*$-algebras. We show that every invariant multilinear CP
map is automatically symmetric and completely bounded. Possibly these results
are unknown in the literature (see \cite{Heo 00,Heo,HJ 2019}). Motivated from
quantum algorithm simulation \cite{BSD} we introduce multilinear version of
invariant block CP map $ \varphi=[\varphi_{ij}] : M_{n}(\A)^k \to
M_n(\mathcal{B({H})}).$ Then we derive that each $\varphi_{ij}$ can be dilated
to a common commutative tuple of$*$-homomorphisms. As a natural appeal, the
suitable notion of minimality has been identified within this framework. A
special case of our result recovers a finer version of J. Heo's Stinespring
type dilation theorem of \cite{Heo}, and A. Kaplan's Stinespring type dilation
theorem \cite{AK89}. As an application, we show Russo-Dye type theorem for
invariant multilinear completely positive maps. Finally, using minimal
Stinespring dilation we obtain Radon Nikodym theorem in this setup.
