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, ℓ1(S)∗∗ 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, ℓ1(S)∗∗ 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.
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