Derivations which are inner as completely bounded maps

TL;DR

This paper investigates when derivations from certain tensor product maps on C*-algebras are inner, establishing conditions for innerness and providing examples of outer derivations.

Contribution

It characterizes when derivations are inner for specific classes of C*-algebras and presents an example of an algebra with outer elementary derivations.

Findings

Derivations are inner if A is prime.

Derivations are inner if A is quasicentral with Hausdorff primitive spectrum.

An example of a C*-algebra with outer elementary derivations is provided.

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.