A Characterization of Polynomially Convex Sets in Banach Spaces

2017/05/15 by Mortaza Abtahi, Abtahi, Mortaza, Sara Farhangi +1 · 1 citation
Mathematics · #46J10 #46J20 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46E50 #Secondary 32E20

paper · pdf · doi:10.48550/arxiv.1705.06589

openalex publication_date 2017/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be a Banach space and \X be the closed unit ball of the dual space E^*. For a compact set K in E, we prove that K is polynomially convex in E if and only if there exist a unital commutative Banach algebra A and a continuous function f:\X→ A such that (1) A is generated by f(\X), (2) the character space of A is homeomorphic to K, and (3) K=\vsp(f) the joint spectrum of f. In case E=\c(X), where X is a compact Hausdorff space, we will see that \X can be replaced by X.

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