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arXiv · 1705.06589

A Characterization of Polynomially Convex Sets in Banach Spaces

Mortaza Abtahi·Sara Farhangi
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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 $f:\X\to A$ 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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Mortaza Abtahi, Sara Farhangi. 2017-05-15. A Characterization of Polynomially Convex Sets in Banach Spaces. https://arxiv.org/abs/1705.06589

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