In Chapter 11 of "Complete Normed Algebras" by Bonsall and Duncan, the following definitions are given:
Definition Let $ A $ be a normed algebra. A Bounded Approximate Identity for $ A $ is a bounded net $ \{ e(\lambda) \} $ in $ A $ such that $ e(\lambda)a \overset{\lambda} \to a $ and $ ae(\lambda) \overset{\lambda} \to a $ for each $ a \in A $.
Definition Let $ A $ be a normed algebra and let $ X $ be a normed $ A $-module. A Bounded Approximate Identity in $ A $ for $ X $ is a bounded net $ \{ e(\lambda) \} $ in $ A $ such that $ e(\lambda)x \overset{\lambda} \to x $ and $ xe(\lambda) \overset{\lambda} \to x $ for each $ x \in X $.
My question is, if $ A $ is a normed algebra that has a bounded approximate identity, then must it have a bounded approximate identity for each of its normed $ A $-modules?
Is it true if $ A $ is a Banach algebra and $ X $ is a Banach $ A $-module?