Elementary operators on Hilbert $C^*$-modules

We extend the well-known notion of elementary operators on $C^*$-algebras to Hilbert $C^*$-modules. After providing some basic properties, we generalize Mathieu's theorem for elementary operators on $C^*$-algebras by showing that the completely bounded norm of any elementary operator on a non-zero Hilbert $A$-module agrees with the Haagerup norm of its corresponding tensor if and only if $A$ is a prime $C^*$-algebra.
This is joint work with Ljiljana Arambašić (University of Zagreb).