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

Abstract: 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.