Abstract. The aim of this paper is to connect the results of D. Bakic and B. Guljas about C*-extensions of Hilbert C*-modules with results of D. P. Blecher about Hilbert C*-extensions of operator spaces. In the first 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.
2000 Mathematics Subject Classification. 46L08, 46L07.
Key words and phrases. Hilbert C*-module, operator space, extension, multiplier algebra, completely bounded map.