Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula σl(S) = {(λ(s))s∈s ∈ C[sup]s | {s - λ(s)}s∈s generates a proper left ideal}. Using the Schur lemma and the Gelfand-Mazur theorem we prove that σl(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.
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We define a new class of Banach algebras of holomorphic functions on the unit disc D which contains the algebras studied in [GMR2] and [GW]. To a function G of the class A1(D) nowhere vanishing in D we associate a Banach algebra AnG(D) contained in the disc algebra A(D). We prove that all closed ideals of AnG(D) of at most countable hull are of the standard form.
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