Abstract

Let E be a Banach space and \({\triangle ^{\!*}}\) 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 \(f:{\triangle ^{\!*}}\rightarrow A\) such that (i) A is generated by \(f({\triangle ^{\!*}})\), (ii) the character space of A is homeomorphic to K, and (iii) \(K=\vec {\textsc {sp}}(f)\) the joint spectrum of f. In case \(E=\mathscr {C}(X)\), where X is a compact Hausdorff space, we will see that \({\triangle ^{\!*}}\) can be replaced by X.

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