A Russo-Dye type Theorem and Stinespring representation for invariant block multilinear completely positive maps

Abstract

In this article, we investigate certain basic properties of invariant multilinear CP maps which hold true in the theory of CP maps between two C * -algebras. In some cases, results are affirmative and in some cases results are negative. For instance, we show that every invariant multilinear CP map is automatically symmetric and completely bounded. Surprisingly these results are unknown in the literature (see [12,13,15]). As a negative result, we provide a concrete example of positive multilinear map ϕ : C(X ) 3 → which is not CP. We also prove Russo-Dye type theorem for invariant multilinear positive maps on a commutative domain.Further, we provide a simple and independent proof of Stinespring dilation theorem for block CP maps. This generalizes the work of A. Kaplan [18]. We introduce multilinear version of invariant block CP map ϕ = [ϕ i j ] : M n ( ) k → M n ( ( )). Then we derive that each ϕ i j can be dilated to a common commutative tuple of * -homomorphisms. This result is a generalization of J. Heo's Stinespring type dilation theorems of [11,13]. As a special case of our result recovers a finer version of J. Heo's Stinespring type dilation theorem of [13] as we do not use the additional assumptions symmetric and completely bounded in the hypothesis. As a natural appeal, the suitable notion of minimality has been identified in this framework. As an application, we show Russo-Dye type theorem for invariant multilinear completely positive maps.