On the extendability of some classes of maps on Hilbert $C^*$-modules

In this paper, we show that every completely semi-$ϕ$-map on a submodule of a Hilbert $C^*$-module has a completely semi-$ϕ$-map extension on the whole of module. We also investigate the extendability of $ϕ$-maps and provide examples of $ϕ$-maps which has no $ϕ$-map extension. Finally, we introduce a category of Hilbert $C^*$-module and determine injective objects in this category.

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