The Bochner–Schoenberg–Eberlein (BSE)-property on commutative Banach algebras is a property related to multiplier algebras of Banach algebras. In this paper, we answer the problem (12) raised by Javanshiri and Nemati in [Amalgamated duplication of the Banach algebra $𝔄$ along a $𝔄$-bimodule ${𝒜}$, J. Algebra Appl. 17(9) (2018) 1850169-1–1850169-21]. In this paper, under certain conditions, we show that the amalgamated Banach algebra ${𝒜}⋊𝔄$ is BSE-algebra if and only if ${𝒜}$ and $𝔄$ are BSE-algebras.

中文翻译：

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巴拿赫代数合并复制的 Bochner-Schoenberg-Eberlein 属性

交换 Banach 代数上的 Bochner-Schoenberg-Eberlein (BSE)-性质是与 Banach 代数的乘数代数相关的性质。在本文中，我们回答了 Javanshiri 和 Nemati 在 [Amalgamated duplication of the Banach 代数中提出的问题 (12)$𝔄$沿着一个$𝔄$-双模块${𝒜}$，J.代数应用。 17 (9) (2018) 1850169-1–1850169-21]。在本文中，在特定条件下，我们证明了合并的 Banach 代数${𝒜}⋊𝔄$是 BSE 代数当且仅当${𝒜}$和$𝔄$是 BSE 代数。