Polynomial identification in uniform and operator algebras
The authors establish various criteria for the equality of two elements in some unital Banach algebras. The first part of the paper is devoted to uniform algebras. Let \(\mathcal A\) be a uniform algebra on a compact Hausdorff space \(K\), and let \(f,g \) be two elements of \(\mathcal A\). Among other results, the authors show that, if there exist \(\alpha, \beta \in \mathbb C \setminus \{ 0 \} \) and \(\gamma \in \mathbb C\) such that \( \| \alpha f + \beta h + \gamma \| = \| \alpha g + \beta h + \gamma \|\) for all \(\mathbb C\)-peaking functions \(h \in \mathcal A\), then \(f=g\). Recall that a nonzero element \(h \in \mathcal A\) is called a \(\mathbb C\)-peaking function if its peripheral spectrum is a singleton. In the second part, the authors consider standard operator algebras. Let \(\mathcal A\) be a unital standard operator algebra on a Banach space \(X\) and let \(A, B\) be in \( \mathcal A\). The authors show that, if there exist \(\alpha, \beta \in \mathbb C \setminus \{ 0 \} \) and \(\gamma \in \mathbb C\) such that \( \rho (\alpha A+ \beta T+ \gamma)= \rho (\alpha B+ \beta T+ \gamma)\) for all \(T \in \mathcal A\), where \(\rho (\cdot)\) denotes the spectral radius, then \(A=B\). Other identification criteria are given.
scientific article; zbMATH DE number 5818627