Recent zbMATH articles in MSC 03E35https://zbmath.org/atom/cc/03E352021-06-15T18:09:00+00:00WerkzeugOn maximal ideals which are also minimal prime ideals in certain Banach rings.https://zbmath.org/1460.460722021-06-15T18:09:00+00:00"Mihara, Tomoki"https://zbmath.org/authors/?q=ai:mihara.tomokiSummary: We study the existence of a maximal ideal which is also a minimal prime ideal in Banach rings in a wide class containing the Banach algebra \(\text{C}_{\text{bd}} (X,k)\) of bounded continuous functions \(X \to k\) for a topological space \(X\) and a Banach field \(k\) with a mild condition, the quotient of \(\text{C}_{\text{bd}} (X,k)\) by the closed ideal \(\text{C}_0(X,k)\) of functions vanishing at infinity, the bounded direct product \(\prod_{\lambda \in \Lambda} \kappa_{\lambda}\) of a family \(\kappa = (\kappa_{\lambda})_{\lambda \in \Lambda}\) of Banach fields with a mild condition, and the quotient of \(\prod_{\lambda \in \Lambda} \kappa_{\lambda}\) by the completed direct sum \(\widehat \bigoplus_{\lambda \in \Lambda} \kappa_{\lambda}\). We describe the maximal spectrum and the Berkovich spectrum of such Banach rings, and generalise the classical result on the relation between the existence of such a maximal ideal of the Banach \(\mathbb{R}\)-algebra \(\text{C}_{\text{bd}} (\mathbb{N},\mathbb{R})/\text{C}_0(\mathbb{N}, \mathbb{R})\) and the existence of a $P$-point in \(\beta \mathbb{N} \setminus \mathbb{N}\).