An attractive and fairly large class of completely bounded linear maps on C*-algebras are elementary operators, that is maps that can be expressed as a finite sum of two-sided multiplications $x \mapsto axb$. Elementary operators provide ways to study the structure of C*-algebras and they also play an important role in modern quantum information theory.

In this talk, we extend the 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 based on joint work with Ljiljana Arambasic (University of Zagreb)