Joint spectra of generators in topological algebras (Q925349)

The author considers a complex topological algebra \(\mathcal A\) and a family of proper closed ideals \(\mathcal U\) in \(\mathcal A\). Fixing a nonempty subset \(S\subset\mathcal A\), a joint spectrum of \(S\) with respect to \(\mathcal U\) may be defined as \(\sigma_{\mathcal U}(S)=\{(\lambda_s)_{s\in S} \in{\mathbb C}^S: \exists I\in{\mathcal U}\) \(\forall s\in S\), \((s-\lambda_s)\in I\}\). It is shown that for a subset \(S\) generating the algebra \(\mathcal A\), the spectrum \(\sigma_{\mathcal U}(S)\) can be identified with the set of continuous multiplicative functionals \(f\) on \(\mathcal A\) such that ker\,\(f\in\mathcal U\).