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

Abstract

Abstract. The aim of this paper is to connect the results of D. Bakić and B. Guljaš 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 defined in [5].