Operators, Spaces, Algebras, Modules 2010

Morphisms out of a split extension of a Hilbert C*-module

Let $W$ be a split extension of a Hilbert $C^*$-module $V$ by a Hilbert $C^*$-module $Z$. Like in the case of $C^*$-algebras (well known theorem of T. A. Loring), every morphism out of $W$, more precisely from $W$ to an arbitrary Hilbert $C^*$-module $U$, is in a bijective correspodence with a pair of morphisms from $V$ and $Z$, respectively, into $U$ which satisfy certain conditions. It turns out that, besides the generalization of the Loring's condition, an additional condition has to be posed.