#### Glasnik Matematicki, Vol. 39, No.2 (2004), 313-326.

### A NOTE ON EXTENSIONS OF HILBERT
C*-MODULES AND THEIR MORPHISMS

### Franka Miriam Bruckler

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia

*e-mail:* `bruckler@math.hr`

**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.

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