On Pseudo-contractibility and Ultra Central Approximate Identity of Some Semigroup Algebras

Abstract

In this paper, we investigate the existence of an ultra central approximate identity for a Banach algebra A and its second dual A∗∗. Also, we prove that for a left cancellative regular semigroup S, ℓ1S∗∗ has an ultra central approximate identity if and only if S is a group. As an application, we show that for a left cancellative semigroup S, ℓ1S∗∗ is pseudo-contractible if and only if S is a finite group. We also study this property for φ-Lau product Banach algebras and the module extension Banach algebras.