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The aim of this paper is to connect the results of D. Baki c and B. Gulja s about C -extensions of Hilbert C -modules with results of D.P. Blecher about Hilbert C -extensions of operator spaces. In the rst part, we give conditions on a completely bounded linear operator between Hilbert C -modules for the possibility of extending the operator to a "corner-preserving" C -morphism of the corresponding linking-algebras (or, equivalently, for the operator being a Hilbert C -morphism). The second part provides an order preserving bijection between the sets of C - extensions of a Hilbert C -module and its Hilbert C -extensions, the latter being a generalized version of Blecher's Hilbert C -extensions of operator spaces dened in (5).

A note on extensions of Hilbert C*-modules and their morphisms