Russo-Dye type Theorem, Stinespring representation,and Radon Nikodym dervative for invariant block multilinear completely positive maps

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.