APPROXIMATELY LOCAL DERIVATIONS

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