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 [12, 13, 16]), Heo and Joiţa (Linear Multilinear Algebra 67 (2019) 121–140). Motivated from quantum algorithm simulation as reported by Bansal et al. [7], we introduce multilinear version of invariant block CP map \( \varphi =[\varphi _{ij}]: M_{n}({\mathcal {A}})^k \rightarrow M_n(\mathcal {B({H})}),\) where \({\mathcal {A}}\) is a \(C^*\) -algebra, and \(\mathcal {B(H)}\) is the set of all bounded linear operators on a Hilbert space \({\mathcal {H}}.\) 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 Heo’s Stinespring type dilation theorem of [13] and Kaplan’s Stinespring type dilation theorem [20]. As an application, we show Russo–Dye type theorem for invariant multilinear completely positive maps. Finally, using minimal Stinespring dilation we obtain Radon–Nikodým theorem in this setup. Our result includes as a special case the Radon–Nikodým theorem of Heo [13] and the Radon–Nikodým theorem of Joiţa [19]
