Abstract
Let X be a nonempty set, A be a commutative Banach algebra, and 1≤p<∞. In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of ℓpX,A. Specifically, we establish that ℓpX,A possesses the BSE property if and only if X is finite and A is BSE. Additionally, we investigate the BSE module property on Banach ℓpX,A-modules and demonstrate that a Banach space ℓpX,Y serves as a BSE Banach ℓpX,A-module if and only if X is finite and Y represents a BSE Banach A-module.
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