We introduce completely semi-ϕ-maps on Hilbert C * -modules as a generalization of ϕ-maps. This class of maps provides examples of CP-extendable maps which are not CP-Hextendable, in Skeide-Sumesh's sense. Using the CP-extendability of completely semi-ϕ-maps, we give a representation theorem, similar to Stinespring's representation theorem, for this class of maps which can be considered as strengthened and generalized form of Asadi's and Bhat-Ramesh-Sumesh's analogues of Stinespring representation theorem for ϕ-maps. We also define an order relation on the set of all completely semi-ϕ-maps and establish a Radon-Nikodym type theorem for this class of maps in terms of their representations.

We provide a characterization for operator-valued completely bounded linear maps on Hilbert C *-modules in terms of ϕ-maps. Also, we show that for every operator-valued completely positive map ϕ on a C *-algebra A , there is a unique (up to multiplication by a unitary operator) non-degenerate ϕ-map on each Hilbert A-module.

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