Derivations which are inner as completely bounded maps

Abstract

We consider derivations from the image of the canonical contraction ${\theta }_A$ from the Haagerup tensor product of a C*-algebra A with itself to the space of completely bounded maps on A. We show that such derivations are necessarily inner if A is prime or if A is quasicentral with Hausdorff primitive spectrum. We also provide an example of a C*-algebra which has outer elementary derivations.

15 pages