Approximate identities for left normed modules in normed algebras

Abstract

We introduce and study some extended notions of approximate identities. It is shown that a normed algebra has a bounded left approximate identity (respectively, bounded two-sided approximate identity) for a left normed -module X if and only if has bounded left approximate units (respectively, bounded two-sided approximate units) for X . Similar results in the situation of bounded right approximate identities for left normed modules are also established. As a consequence, we give an affirmative answer to a question of Doran and Wichmann.