Schur Lemma and the Spectral Mapping Formula
Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula = generates a proper left idealUsing the Schur lemma and the Gelfand-Mazur theorem we prove that 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.