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

Wawrzyńczyk, Antoni

Bull. Pol. Acad. Sci., Math. 55, No. 1, 63-69 (2007).

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.

MSC:

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

Keywords:

Waelbroeck algebra; joint spectrum; spectral mapping formula