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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.
We generalize to some classes of ultradifferentiable jets or functions the classical Łojasiewicz Division Theorem and Glaeser Composition Theorem. The proof uses the desingularization results by Hironaka, Bierstone and Milman.
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