The spectra of algebras of Lorch analytic mappings
Let \(E\) be a commutative Banach algebra with a unit element \(e\) and \(U\) be a connected open subset of \(E\). A function \(f: U\to E\) is said to be \((L)\)-analytic if for each \(a\) in \(U\) there is a unique sequence \((a_n)_n\) in \(E\) such that \(f(z)=\sum_{n=0}^\infty a_n(z-a)^n\) for all \(z\) in some neighbourhood of \(a\). The authors prove that \({\mathcal H}_L(E)\), the algebra of all \((L)\)-analytic functions \(f: E\to E\), is a commutative Fréchet algebra with unit when endowed with the topology of uniform convergence on bounded subsets of \(E\). For an algebra \({\mathcal A}\) let \({\mathcal M}({\mathcal A})\) denote the spectrum of \({\mathcal A}\). It is shown that the mapping \(\delta: {\mathcal M}({\mathcal A})\times {\mathbb{C}}\to {\mathcal M}({\mathcal H}_L(E))\), \(\delta(\varphi, \lambda)(f)=\varphi(f(\lambda e))\), is injective, onto and has a continuous inverse. Hence \({\mathcal H}_L(E)\) is semi-simple if and only if \(E\) is semi-simple. The authors use \({\mathcal A}_L(B_E)\) to denote the algebra of all \((L)\)-analytic functions \(f: B_E\to E\) which have continuous extension to \(\overline{B_E}\) and \({\mathcal H}^\infty_L(\text{int}B_E)\) to denote the algebra of all bounded \((L)\)-analytic functions \(f: B_E\to E\). It is shown that \({\mathcal M}({\mathcal A}_L(B_E))\) is homeomorphic to \({\mathcal M}(E)\times \overline\Delta\) and there is a continuous surjection from \({\mathcal M}({\mathcal H}^\infty_L(\text{int}B_E))\) onto \({\mathcal M}(E)\times \overline\Delta\) which is injective on \(\delta({\mathcal M}(E)\times \Delta)\). A necessary and sufficient condition to ensure that \(\overline{\delta({\mathcal M}(E)\times \Delta)}={\mathcal M}({\mathcal H}^\infty_L(\text{int}B_E))\) is provided by the authors.
scientific article; zbMATH DE number 5706249