Hilbert \(C^*\)-modules as a subcategory of operator systems and injectivity
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In this paper, we study a category whose objects are Hilbert \(C^*\)-modules and whose morphisms are completely semi-\(\phi \)-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-\(\phi \)-maps on Hilbert \(C^*\)-modules, leading to an analog of the Arveson’s extension theorem for completely semi-\(\phi \)-maps (in contrast with \(\phi \)-maps). This theorem together with previous results suggest that the completely semi-\(\phi \)-maps are proper generalizations of the completely positive maps.
KeywordsHilbert \(C^*\)-modules Extension theorems Injective objects Completely positive maps Completely semi-\(\phi \)-maps
Mathematics Subject ClassificationPrimary 46L08 Secondary 46L07
The research of the first author was in part supported by a Grant from IPM (No. 94470046).
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