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

中文翻译：

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向量值$$ \ ell ^ p $$ℓp-空间的Bochner–Schoenberg–Eberlein属性

令X为非空集，\（{\ mathcal {A}} \）为可交换Banach代数，而\（1 \ le p <\ infty \）。在本文中，我们建立了\（\ ell ^ p（X，\ mathcal {A}）\）的一些基本属性，它们继承自\（{\ mathcal {A}} \）。特别是，我们表征\（\ ell ^ p（X，\ mathcal {A}）\）的Gelfand空间，用\（\ Delta（\ ell ^ p（X，{\ mathcal {A}）}）表示\）。我们主要研究Banach代数\（\ ell ^ p（X，\ mathcal {A}）\）的BSE性质。实际上，我们证明\（\ ell ^ p（X，\ mathcal {A}）\）是BSE代数，当且仅当X是有限的且\（\ mathcal {A} \）是BSE代数。此外，在\（\ mathcal {A} \）是单位的情况下，我们表明对于任何自然数n，\（\ Delta（\ ell ^ p（X，{\ mathcal {A} }））\）是n-BSE函数。但是，通过一个示例，我们表明\（\ Delta（\ ell ^ p（X，{\ mathcal {A}}））\）上存在一些连续的有界函数，而不是BSE。最后，我们证明如果\（\ ell ^ 1（X，{\ mathcal {A}}）\）是BSE范数代数，则\（\ mathcal {A} \）就是BSE范数代数。每当\（\ mathcal {A} \）是一个最高范数代数时，我们还证明了该陈述的相反含义。