# A Characterization of Polynomially Convex Sets in Banach Spaces

## TL;DR

This paper characterizes polynomially convex sets in Banach spaces by linking them to spectra of functions into Banach algebras, providing a new functional-analytic description of such sets.

## Contribution

It establishes an equivalence between polynomial convexity of compact sets in Banach spaces and the existence of certain Banach algebra representations, extending classical results.

## Key findings

- Polynomially convex sets correspond to spectra of functions into Banach algebras.
- Characterization holds for general Banach spaces and simplifies in the case of $E= C(X)$.
- Provides a functional-analytic framework for understanding polynomial convexity in infinite-dimensional spaces.

## 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 $X$.