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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 of such CP-extendable maps whose 22-corner is a homomorphism, the CP-H-extendable maps. We show that they coincide with the maps considered by Asadi [Asa09], by Bhat, Ramesh, and Sumesh [BRS12], and by Skeide [Ske10]. We also give an intrinsic characterization that generalizes the characterization by Abbaspour and Skeide [AbSk07] of homomorphicly 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. As an application, we examine semigroups of CP-H-extendable maps, so-called CPH-semigroups, and illustrate their relation with a sort of generalized dilation of CP-semigroups, CPH-dilations.