Schur lemma and the spectral mapping formula. (English) Zbl 1118.46045

Summary: Let \(B\) be a complex topological unital algebra. The left joint spectrum of a set \(S\subset B\) is defined by the formula
\[ \sigma_l(S)=\{(\lambda(s))_{s\in S}\in\mathbb C^S \mid \{s-\lambda(s)\}_{s\in S}\text{ generates a proper left ideal}\}. \]
Using the Schur lemma and the Gelfand-Mazur theorem, we prove that \(\sigma_l(S)\) has the spectral mapping property for sets \(S\) of pairwise commuting elements if \(B\) is an \(m\)-convex algebra with all maximal left ideals closed, or if \(B\) is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.


46H10 Ideals and subalgebras
46H30 Functional calculus in topological algebras
46H15 Representations of topological algebras
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