Positivity DOI 10.1007/s11117-017-0530-9 Positivity Hilbert C * -modules as a subcategory of operator systems and injectivity Mohammad B. Asadi 1,2 · Reza Behmani 3 · Ali R. Medghalchi 3 · Hamed Nikpey 4 Received: 31 July 2017 / Accepted: 19 September 2017 © Springer International Publishing AG 2017 Abstract In this paper, we study a category whose objects are Hilbert C ∗ -modules and whose morphisms are completely semi-φ-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-φ- maps on Hilbert C ∗ -modules, leading to an analog of the Arveson’s extension theorem for completely semi-φ-maps (in contrast with φ-maps). This theorem together with previous results suggest that the completely semi-φ-maps are proper generalizations of the completely positive maps. Keywords Hilbert C ∗ -modules · Extension theorems · Injective objects · Completely positive maps · Completely semi-φ-maps Mathematics Subject Classification Primary 46L08; Secondary 46L07 B Mohammad B. Asadi mb.asadi@khayam.ut.ac.ir Reza Behmani reza.behmani@gmail.com Ali R. Medghalchi a_medghalchi@khu.ac.ir Hamed Nikpey hamednikpey@gmail.com 1 School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran 2 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran 3 Department of Mathematics, Kharazmi University, 50, Taleghani Ave., Tehran 15618, Iran 4 Department of Mathematics, Shahid Rajaee Teacher Training University, Tehran 16785-136, Iran