The BSE-property for vector-valued L^p-algebras

2022/06/14 by Fatemeh Abtahi, Abtahi, Fatemeh, Mitra Amiri +3
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2206.07123

openalex publication_date 2022/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathcal A be a separable Banach algebra, G be a locally compact Hausdorff group and 1< p<∞. In this paper, we first provide a necessary and sufficient condition, for which L^p(G,\mathcal A) is a Banach algebra, under convolution product. Then we characterize the character space of L^p(G,\mathcal A), in the case where \mathcal A is commutative and G is abelian. Moreover, we investigate the BSE-property for L^p(G,\mathcal A) and prove that L^p(G,\mathcal A) is a BSE-algebra if and only if \mathcal A is a BSE-algebra and G is finite. Finally, we study the BSE-norm property for L^p(G,\mathcal A) and show that if L^p(G,\mathcal A) is a BSE-norm algebra then \mathcal A is so. We prove the converse of this statement for the case where G is finite and \mathcal A is unital.

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