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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.
Antoni Wawrzyńczyk. "Schur Lemma and the Spectral Mapping Formula." Bulletin of the Polish Academy of Sciences. Mathematics 55.1 (2007): 63-69. <http://eudml.org/doc/280516>.
@article{AntoniWawrzyńczyk2007, abstract = {Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula
$σ_l(S)$ = $(λ(s))_\{s∈ S\} ∈ ℂ^S | \{s-λ(s)\}_\{s∈S\}$ generates a proper left ideal$.
$Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_l(S)$ 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.}, author = {Antoni Wawrzyńczyk}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, keywords = {Waelbroeck algebra; joint spectrum; spectral mapping formula}, language = {eng}, number = {1}, pages = {63-69}, title = {Schur Lemma and the Spectral Mapping Formula}, url = {http://eudml.org/doc/280516}, volume = {55}, year = {2007}, }
TY - JOUR AU - Antoni Wawrzyńczyk TI - Schur Lemma and the Spectral Mapping Formula JO - Bulletin of the Polish Academy of Sciences. Mathematics PY - 2007 VL - 55 IS - 1 SP - 63 EP - 69 AB - Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula
$σ_l(S)$ = $(λ(s))_{s∈ S} ∈ ℂ^S | {s-λ(s)}_{s∈S}$ generates a proper left ideal$.
$Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_l(S)$ 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. LA - eng KW - Waelbroeck algebra; joint spectrum; spectral mapping formula UR - http://eudml.org/doc/280516 ER -