On maximal ideals which are also minimal prime ideals in certain Banach rings
Volume 250 / 2020
Abstract
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 ${\rm C}_{{\rm bd}} (X,k)$ of bounded continuous functions $X \to k$ for a topological space $X$ and a Banach field $k$ with a mild condition, the quotient of ${\rm C}_{{\rm bd}} (X,k)$ by the closed ideal ${\rm C} _0(X,k)$ 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\oplus_{\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 ${\rm C}_{{\rm bd}} (\mathbb N ,\mathbb R )/{\rm C} _0(\mathbb N ,\mathbb R )$ and the existence of a P-point in $\beta \mathbb N \setminus \mathbb N $.
