Peripheral spectrum for \(A \times B\) (Q2815607)

Let \(A\) be a function algebra on a compact Hausdorff space \(X\). The authors study the maximum modulus set of an element \(h=(f,g)\in A\times B\) and its peripheral spectrum NEWLINE\[NEWLINE\sigma_\pi(h)=\sigma(h)\cap \{z\in \mathbb C:\| h\|=|z|\}NEWLINE\]NEWLINE within the Cartesian product \(A\times B\) of two function algebras and where \(\| h\|=\max\{\| f\|_\infty, \| g\|_\infty\}\). Some observations on the set of peaking functions for \(A\times B\) are given, too.