Approximately Local Derivations

Certain linear operators from a Banach algebra A into a BanachA-bimodule X, which are called approximately local derivations,are studied. It is shown that when A is a C^*-algebra, a Banachalgebra generated by idempotents, a semisimple annihilator Banachalgebra, or the group algebra of a SIN or a totally disconnectedgroup, bounded approximately local derivations from A into Xare derivations. This, in particular, extends a result of B.E. Johnson that ‘local derivations on C^*-algebras arederivations’ and provides an alternative proof of it.