Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula ${\sigma }_l\left(S\right)$ = ${\left(\lambda \left(s\right)\right)}_{s\in S}\in {ℂ}^S|{s-\lambda \left(s\right)}_{s\in S}$ generates a proper left ideal$.$Using the Schur lemma and the Gelfand-Mazur theorem we prove that ${\sigma }_l\left(S\right)$ 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.