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.