The spectral mapping formula in Arens-Michael-Fréchet algebras
type: Item
instance of: scholarly article
title: The spectral mapping formula in Arens-Michael-Fréchet algebras
zbMATH Open document ID: 1226.46048
author: Armando Velázquez González
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2011-05-18T00:00:00Z
review text: Let \(A\) be a Arens-Michael-Fréchet (or locally \(m\)-convex Fréchet) algebra with unit \(e\). The main result of the present paper is the following: If for any \((k+n)\)-tuple of mutually commuting elements \(a_1,\dots,a_k,b_1,\dots,b_n\in A\), the left (right) ideal in \(A\) generated by \(a_1,\dots,a_k\) is proper in \(A\), then there exists \((\lambda _1,\dots,\lambda _n)\in\mathbb{C}^n\) such that the left (respectively right) ideal in \(A\) generated by \(a_1,\dots,a_k,b_1-\lambda _1e,\dots, b_n-\lambda_ne\) is proper in \(A\). Using this result, it is shown that the left (right) joint spectrum of a \(k\)-tuple of mutually commuting elements \(a_1,\dots, a_k\in A\) is not empty and it satisfies the spectral mapping formula.
reviewed by: Mati Abel
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 5896970
zbMATH Keywords: locally \(m\)-convex Fréchet algebra
zbMATH Keywords: left (right) joint spectrum
zbMATH Keywords: spectral mapping formula
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5896970