Schur Lemma and the Spectral Mapping Formula

Antoni Wawrzyńczyk¹

¹ Departamento de Matemáticas Universidad Autónoma Metropolitana – Iztapalapa Av. San Rafael Atlixco 186, col. Vicentina AP 55-534 09 340 México, D.F., Mexico

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) no. 1, pp. 63-69.

Résumé

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.

Détail

DOI : 10.4064/ba55-1-7

Keywords: complex topological unital algebra joint spectrum set subset defined formula sigma lambda mathbb mid s lambda hbox generates proper ideal using schur lemma gelfand mazur theorem prove sigma has spectral mapping property sets pairwise commuting elements m convex algebra maximal ideals closed locally convex waelbroeck algebra right ideal version result valid