On the extendability of some classes of maps on Hilbert $C^*$-modules
  In this paper, we show that every completely semi-$\phi$-map on a submodule of a Hilbert $C^*$-module has a completely semi-$\phi$-map extension on the whole of module. We also investigate the extendability of $\phi$-maps and provide examples of $\phi$-maps which has no $\phi$-map extension. Finally, we introduce a category of Hilbert $C^*$-module and determine injective objects in this category. 
Mohammad B. Asadi, Reza Behmani, Ali R. Medghalchi and Hamed Nikpey
math.OA