CP-H-extendable maps between Hilbert modules and CPH-semigroups

One may ask which maps between Hilbert modules allow for a completely positive extension to a map acting block-wise between the associated (extended) linking algebras. In these notes we investigate in particular those CP-extendable maps where the 22-corner of the extension can be chosen to be a homomorphism, the CP-H-extendable maps. We show that they coincide with the maps considered by Asadi [4], by Bhat, Ramesh, and Sumesh [9], and by Skeide [28]. We also give an intrinsic characterization that generalizes the characterization by Abbaspour Tabadkan and Skeide [1] of homomorphically extendable maps as those which are ternary homomorphisms. For general strictly CP-extendable maps we give a factorization theorem that generalizes those of Asadi, of Bhat, Ramesh, and Sumesh, and of Skeide for CP-H-extendable maps. This theorem may be viewed as a unification of the representation theory of the algebra of adjointable operators and the KSGNS-construction. Then, we examine semigroups of CP-H-extendable maps, so-called CPH-semigroups. As an application, we illustrate their relation with a new sort of generalized dilation of CP-semigroups, CPH-dilations.