Operators, Spaces, Algebras, Modules 2010

Derivations, completely bounded maps and the ideal structure of C*-algebras

We consider derivations in the image of the canonical contraction $\theta_A$ from the Haagerup tensor product of a quasicentral $C^*$-algebra $A$ with itself to the space of completely bounded maps on $A$. We show that such derivations are necessarily inner if every Glimm ideal of $A$ is prime. We also provide an example of a $C^*$-algebra which has an outer derivation implemented by an elementary operator.