Abstract
Abstract Let 𝓐 be a Banach algebra and let 𝓜 be a unital Banach algebra. For a homomorphism Φ from 𝓐 into 𝓜, we consider 𝓜 as a Banach right 𝓐-module and investigate when 𝓜 is a retract of 𝓐 with respect to Φ. We also give characterizations of admitting vector-valued invariant Φ-means in terms of projectivity and injectivity. Finally, we apply these results to abstract Segal algebras.
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