APPROXIMATELY LOCAL DERIVATIONS

Abstract

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 ${\rm C^*}$-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 ${\rm C^*}$-algebras are derivations’ and provides an alternative proof of it.