Schur Lemma and the Spectral Mapping Formula
Volume 55 / 2007
Bulletin Polish Acad. Sci. Math. 55 (2007), 63-69
MSC: Primary 46H10; Secondary 46H15, 46H30.
DOI: 10.4064/ba55-1-7
Abstract
Let $B$ be a complex topological unital algebra. The left joint spectrum of a set $S\subset B$ is defined by the formula $$ \sigma_l(S)=\{(\lambda(s))_{s\in S}\in\mathbb C^S\mid \{s-\lambda(s)\}_{s\in S} \hbox{ generates a proper left ideal}\}. $$ 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
(i) $B$ is an m-convex algebra with all maximal left ideals closed, or
(ii) $B$ is a locally convex Waelbroeck algebra.
The right ideal version of this result is also valid.
