A commutative Banach algebra which factorizes but has no approximate units

Abstract

It is well known that any Banach algebra having bounded approximate units factorizes. For some time it was not clear if, conversely, factorization implied the existence of bounded approximate units. This was disproved by Paschke [3], but the problem remained open for commutative Banach algebras. We give an example of a commutative semisimple Banach algebra which factorizes but has not even unbounded approximate units.