Topological radical of Banach module

TL;DR

This paper introduces the topological radical of a Banach module, a new concept that generalizes the radical of modules over rings within a functional analytic framework, with similar categorical properties.

Contribution

It defines the topological radical of Banach modules and explores its properties, providing a functional analytic analogue to algebraic radical concepts.

Findings

The topological radical is characterized as the intersection of ranges of maximal contractive monomorphisms.

It can also be described as the union of ranges of small morphisms.

The concept retains key categorical properties similar to algebraic radicals.

Abstract

We introduce the concept of topological radical of a Banach module. This closed submodule have two description: the as the intersection of ranges of maximal contractive monomorphism (from outside) and as the union of ranges of small morphisms (from inside). This concept is a functional analytic analogue for radical of module over a unital ring and has the similar categorical properties.