A uniqueness result concerning Schur ideals

A subset of the set of all positive semi-definite matrices of a given size which is invariant under Schur (componentwise) multiplication by an arbitrary positive semi-definite matrix is said to be a Schur ideal. A subset of $k$-dimensional complex space is said to be $hyperconvex$ if it arises as the set of possible values $(w_{1}, \dots, w_{k}) = (f(\alpha_{1}), \dots, f(\alpha_{k}))$arising from restricting contractive elements $f$ from some uniform algebra $A$ to a finite set $\{ \alpha_{1}, \dots, \alpha_{k} \}$ in the domain. When the uniform algebra is the disk algebra, the hyperconvex set is said to be a Pick body. Motivated by the classical Pick interpolation theorem, Paulsen has introduced a natural notion of duality between Schur ideals and hyperconvex sets. By using some recently developed results in operator algebras (matricial Schur ideals), we show that each Pick body has a unique affiliated Schur ideal.