A Characterization of Polynomially Convex Sets in Banach Spaces

Let $E$ be a Banach space and $\text{X}$ be the closed unit ball of the dual space $E^{\ast }$. 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 $f:\text{X}\to A$ such that (1) $A$ is generated by $f(\text{X})$, (2) the character space of $A$ is homeomorphic to $K$, and (3) $K=\text{vsp}(f)$ the joint spectrum of $f$. In case $E=\stackrel{¸}{(}X)$, where $X$ is a compact Hausdorff space, we will see that $\text{X}$ can be replaced by $X$.