A Radon-Nikodým theorem for local completely positive invariant multilinear maps

Abstract

In this article, we introduce local completely positive $k$-linear maps between locally $C^{\ast }$-algebras and obtain Stinespring type representation by adopting the notion of "invariance" defined by J. Heo for $k$-linear maps between $C^{\ast }$-algebras. Also, we supply the minimality condition to make certain that minimal representation is unique up to unitary equivalence. As a consequence, we prove Radon-Nikodým theorem for unbounded operator-valued local completely positive invariant $k$-linear maps. The obtained Radon-Nikodým derivative is a positive contraction on some Hilbert space with several reducing subspaces.

22 pages (to appear in Linear and Multilinear Algebra)