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Peripheral spectrum for \(A \times B\). (English) Zbl 1355.46046

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 \[ \sigma_\pi(h)=\sigma(h)\cap \{z\in \mathbb C:\| h\|=|z|\} \] 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.

MSC:

46J10 Banach algebras of continuous functions, function algebras
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