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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?

LMW
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  • Are you sure your second definition (for modules) is correct? I don't have the book with me at the moment, but modules are typically one-sided... – zipirovich Aug 19 '18 at 04:12
  • Perhaps my definition is wrong. I assumed that the definition of a left bounded approximate identity in $ A $ for a left normed $ A $-module and a right bounded approximate in $ A $ for a right normed $ A $-module would naturally extend to a "bounded approximate identity in $ A $" for a normed $ A $-bimodule/module. Is this wrong? – LMW Aug 19 '18 at 05:39
  • Of course, we can speak of bimodules, but you didn't say so -- you only said that $X$ is a module. And even then, i.e. if $X$ is an $A$-$A$-bimodule, things may work differently from the two sides. – zipirovich Aug 19 '18 at 05:43
  • Hm, I thought the terms "module" and "bimodule" were synonymous. – LMW Aug 19 '18 at 06:04

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