The Bochner–Schoenberg–Eberlein Property for Vector-Valued ℓp -Spaces

Abstract

Let X be a non-empty set, A be a commutative Banach algebra, and 1 ≤ p< ∞. In this paper, we establish some basic properties of ℓp(X, A) , inherited from A. In particular, we characterize the Gelfand space of ℓp(X, A) , denoted by Δ (ℓp(X, A) ). Mainly, we investigate the BSE property of the Banach algebra ℓp(X, A). In fact, we prove that ℓp(X, A) is a BSE algebra if and only if X is finite and A is a BSE algebra. Furthermore, in the case that A is unital, we show that for any natural number n, all continuous bounded functions on Δ (ℓp(X, A) ) are n-BSE functions. However, through an example, we indicate that there is some continuous bounded function on Δ (ℓp(X, A) ) which is not BSE. Finally, we prove that if ℓ1(X, A) is a BSE-norm algebra, then A is so. We also prove the converse of this statement, whenever A is a supremum norm algebra. © 2020, Springer Nature Switzerland AG.