On the extendability of some classes of maps on Hilbert $C^{\ast }$-modules

Abstract

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