The spectral mapping formula in Arens-Michael-Fréchet algebras
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.
scientific article; zbMATH DE number 5896970